GEOMETRY  AND  FAITH 


OMAS    HILL  D.D. 


UUSB  LIBRARY 


GEOMETRY  AND   FAITH. 


"No  man,  therefore,  can  doubt,  but  toward  the  atteining  of  knowledg 
incomparable,  and  Heavenly  wisdome,  mathematical  speculations,  both 
of  Numbers  and  Magnitudes,  are  means,  aids  and  guides;  ready,  certain 
and  necessary." — John  Dee,  of  London,  1570. 


GEOMETRY  AND  FAITH 


A    SUPPLEMENT 


BY 

THOMAS    HILL 


"The  truths  of  Natural  Religion  are  impressed  In  indelible   characters 
on  every  fragment  of  the  material  world  " 


THIRD  EDITION  GREATLY  ENLARGED 


BOSTON 
LEE    AND    SHEPARD    PUBLISHERS 

NEW  YORK   CHARLES  T.  DILLINGHAM 

1882 


"  This  is  the  chiefe  Glory  of  Geometry,  that  it  loyters  not,  or  employes 
it  self  about  these  inferiour  Machines,  from  whence  it  had  its  Original, 
but  hath  soared  up  into  Heaven,  and  resetled  humane  minds,  (groveling 
before  in  the  dust)  in  Ct--'hstial  Seats." — John  Lteke  and  George  Serle, 
1661. 


COPYRIGHT,  1882, 
BY    LEE   AND    SHEPARD. 


All  rights  reserved. 


CONTENTS. 


Pag*. 

CHAP.    I.     SYMMETRY  IN  SPACE I 

II.     SYMMETRY  IN  TIME 10 

III.  NUMBER 15 

IV.  THE  CALCULUS 20 

V.  APPLIED  MATHEMATICS     .        .        .        .        .24 

VI.     MOTION 27 

VII.     MUSCULAR  ACTION 32 

VIII.  GEOMETRICAL  INSTINCTS     ....         37 

IX.  MOTION  ETERNAL  IN  DURATION  ...    43 

X.  MOTION  OMNIPRESENT  IN  SPACE     .        .         46 

XI.  THE  SPHERE  OF  HUMAN  INFLUENCE  .        .     50 

XII.     MAGNITUDE 54 

XIII.     CHANCE  AND  AVERAGE 64 

XIV.     PHYLLOTAXIS 1\ 

XV.  NUMBER  AND   PROPORTION      ....    90 

XVI.  THE   DEVELOPMENT   OF   FORMS                 .        102 


*  yuannQiav,  *  *  o  drj  ffavpa  ovx 
Oeiov  yavcQov  nv  yiyvono  -ccp  dwapsvm  %vvvou>ir'  *  *  o  8t  6ei6v  -f  eari 


xa<  davpaarb*  roJg  lyxa^o^wat  re  xat  Siavoov^eroic;n  —  The  Epinomis. 


PREFACE. 


"  THE  present  work  lays  no  claim  to  originality."  A  humble 
gleaner  in  the  fields  of  Ma  thesis  and  Theology,  I  offer  only  a  few 
of  the  common  fruits,  well  known  to  those  who  have  more  thoroughly 
surveyed  the  boundaries  of  these  two  domains.  And  I  have  ventured 
to  connect  them  with  the  name  of  the  "  Bridgewater  Treatises,"  not 
because  I  consider  myself  worthy  to  appear  in  company  with  their 
writers,  but  simply  that  I  may  thus  the  more  earnestly  express  my 
admiration  for  the  treatise  of  Babbage. 

WALTHAM,  MASS.,  1849. 


PREFACE  TO   THE   SECOND   EDITION. 

I  have  rewritten  the  greater  part  of  this  work,  and  altered  so  much 
the  expression  and  illustration  of  my  thought,  that  I  might  have  given 
it  a  new  title,  but  for  the  affection  which  twenty -five  years'  familiarity 
has  bred  for  the  old  one. 

PORTLAND,  ME.,  1874. 


Evpoloyrpw,   £(ftr  rov  yog  dei  ovrog  r\   veafjtcrQtxi]  yvmais  B 
'0\xbv  aQct,  w  ytwats,  yv^s  TtQog  d^dsiav  laj  av,  xat  aJiSQ 
qnloaoyov   d/arot'ag  nQog   TO  av<o   ^uv   a  vw  xoaea  ov  dsov 
—  Plato's  Republic,  Book  VII. 


GEOMETRY  AND  FAITH. 


I. 

SYMMETRY  IN   SPACE. 

THE  universe,  actual,  possible  and  impossible,  is  composed  of 
four  elements,  spirit,  matter,  space,  and  time,  which  are  by  no 
alchemy  transmutable  into  each  other.  Many  alchemists  continue, 
even  in  this  closing  half  of  the  nineteenth  century,  to  make  the 
attempt,  and  some  even  flatter  themselves  that  they  are  succeed- 
ing ;  but  the  sturdy  reply  of  human  consciousness  is,  that  the  four 
elements  are  diverse  and  not  transmutable ;  or,  if  any  trans- 
mutation is  possible,  it  must  be  confined  to  this,  that  matter  may, 
in  some  manner,  be  an  effect  of  spirit.  But  to  us,  finite  spirits, 
nothing  more  is  granted  than  the  re-arrangement,  the  partial  con- 
trol of  matter,  not  its  creation.  Matter,  as  we  know  it,  is  dis- 
tinguished by  its  being  the  recipient  and  dispenser  offeree  ;  which 
force,  so  far  as  we  know  it,  is  from  spirit  alone.  This  obedience 
of  matter  to  spirit  gives  justification  to  our  suspicion  that  it  is  the 
creation  of  spirit. 

Space  and  time  are  without  parts,  and  are  indivisible  except  by 
a  mental  act.  This  division  is  suggested  to  us  by  manifest  motion 
in  matter.  Force  shows  itself  in  matter  by  moving  it ;  that  motion 
calls  our  attention  to  the  space  and  time,  within  which  the  motion 

(i) 


2  GEOMETRY   AND    FAITH. 

is  taking  place  ;  and  we  divide  mentally  this  space  and  time,  first 
from  the  remainder  of  the  boundless  contiguities,  secondly  into 
smaller  parts.  Thus  geometry  and  algebra  are  generated,  the 
sciences  which  deal  respectively  with  space  and  time,  those  pure 
entities,  the  relation  of  which  to  the  Infinite  Spirit  we  cannot  com- 
prehend, but  which  we  become  familiar  with  in  the  finite  portion 
embraced  in  our  experience,  in  the  universe  and  its  history. 

In  geometry,  the  mind  imposes  upon  indivisible  space  arbi- 
trary boundaries  of  division,  according  to  arbitrarily  selected  laws 
or  conditions.  These  boundaries  are  of  three  kinds,  surfaces,  lines 
and  points.  The  point  is  a  zero  of  magnitude  in  space,  but  never- 
theless is  not  nothing  ;  which  is  nowhere,  while  the  point  is  some- 
where. This  contradiction  in  terms,  that  a  point  should  have  no 
extension,  and  yet  have  a  position,  is  one  of  those  instances,  in 
which  geometry  abounds,  in  which  the  mind  is  compelled,  by  the 
necessity  of  direct  vision,  to  admit  each  of  two  truths,  which  are  to' 
logic  mutual  contradictories.  The  mathematician  modifies  the  law 
of  non-contradiction  by  confining  it  to  propositions  concerning  finite 
quantities. 

A  lower  form  of  a  zero -of  magnitude  in  space  is  the  line,  which 
is  extended,  at  each  point,  only  in  two  opposite  directions  ;  and  the 
lowest  form  is  the  surface  ;  for  which  there  can,  at  each  point,  be 
drawn  a  line,  such  that  the  surface  extends,  in  every  direction, 
only  perpendicular  to  that  line.  Geometers  define  these  lower 
forms  of  zeroes,  or  boundaries  in  space,  by  the  further  self-contra- 
diction of  imagining  the  movement  of  a  point ;  a  double  contradic- 
tion, since  space  is  itself  incapable  of  motion,  much  more  a  zero 
of  magnitude. 

A  geometrical  line  is  defined  as  the  path  of  a  point,  moving  ac- 
cording to  certain  conditions,  which  always  limit  its  motion,  in 


SYMMETRY   IN   SPACE.  3 

each  of  its  positions,  to  one  of  two  opposite  directions.  Or,  it 
may  be  defined  as  a  continuous  series  of  all  the  points  which  fulfill 
certain  conditions,  among  which  must  be  the  condition  that  each 
point  is  contiguous  only  to  two  others,  one  on  the  opposite  side  to 
the  other.  So  also  a  surface  may  be  defined  as  the  space  in  which 
a  point  moves,  when,  in  each  position  which  it  assumes,  a  straight 
line  may  be  drawn  through  it,  and  its  motion  be  permitted,  in  any 
direction  at  right  angles  to  that  line,  and  in  no  other.  Or,  the 
surface  may  be  defined  as  a  series  of  points,  through  any  one  of 
which  a  straight  line  may  be  drawn,  such  that  all  the  contiguous 
points  lie  in  a  direction  at  right  angles  to  that  line.  To  either  of 
these  definitions  of  a  surface,  we  must  add,  in  order  to  make  a 
geometrical  surface,  some  other  conditions  which  the  points  must 
fulfill. 

When  the  geometer  has  selected  these  conditions  and  would  in- 
vestigate the  form  which  the  points,  so  conditioned,  would  enclose, 
he  is  not  contented  with  the  mere  act  of  reason ;  he  endeavors  to 
bring  imagination  to  his  aid ;  to  make  a  sensible  image  of  the  form. 
If  he  has  been  blind  from  his  birth,  he  imagines  his  fingers  feeling 
out  the  form  ;  otherwise  he  embodies  it  visibly,  as  in  a  drawing, 
or  in  a  model.  If  he  would  convey  a  knowledge  of  it  to  others, 
he  calls  matter  to  his  aid,  and  forces  atoms  of  chalk,  black  lead, 
wood  or  thread,  to  fulfill  approximately  the  conditions  which  his 
geometric  law  imposes  upon  the  series  of  points.  This  drawing, 
or  model,  is  an  expression  of  his  idea,  an  enunciation  of  his  law. 
A  geometrical  figure,  whether  upon  the  blackboard,  or  the  printed 
page,  or  in  a  block  of  wood,  or  a  set  of  stretched  threads,  is  incon- 
trovertible evidence  that  a  geometer  has  been  expressing,  by  this 
means,  a  geometrical  thought. 

The  laws  which  please  the  geometer  most  highly  are  those 


4  GEOMETRY   AND    FAITH. 

which  give  us  symmetrical  figures,  figures  in  which  part  answers 
to  part ;  either  on  opposite  sides  of  one  line  or  one  surface,  or 
about  more  than  one  line  or  surface.  This  taste  is  not  peculiar  to 
the  geometer ;  symmetry  pleases  the  most  savage,  as  it  does  the 
civilized  man;  and  men  whose  whole  ability  lies  in  other  direc- 
tions, as  well  as  the  mathematician.  A  striking  proof  of  the  uni- 
versality of  this  taste  was  shown  in  the  sudden  and  universal  pop- 
ularity attained  by  the  kaleidoscope.  In  a  few  years  that  toy  of 
Brewster  found  its  way  to  every  parlor,  and  the  heart  of  every 
child,  ay,  and  every  man  in  Christendom.  Yet  its  sole  magic 
consists  in  the  symmetry  which  it  imparts  to  a  few  fragments  of 
irregular  form.  But  that  magic  is  sufficient  to  enchant  all  who 
come  within  its  sway.  We  have  never  found  any  one  uninterested 

•/  •/ 

in  an  extempore  kaleidoscope,  made  by  throwing  open  the  piano, 
and  placing  brightly  colored  articles  at  one  end  of  the  folding  lid. 

All  regularity  of  form  is  as  truly  an  expression  of  thought  as  a 
geometrical  diagram  can  be.  The  particles  of  matter  take  the 
form  in  obedience  to  a  force  which  is  acting  according  to  an  intel- 
lectual law,  imposing  conditions  on  its  exercise.  It  does  not  alter 
the  reality  of  this  ultimate  dependence  of  symmetry  upon  thought, 
simply  to  introduce  a  chain  of  secondary  canses,  between  the  origi- 
nal thinking  and  the  final  expression  of  the  thought. 

Many  of  the  geometer's  a  priori  laws  were,  indeed,  first  sug- 
gested by  the  forms  of  nature.  Natural  symmetry  leads  us  to 
investigate,  first,  the  mathematical  law  which  it  embodies ;  then, 
the  mechanical  law  which  embodies  it.  Thus  all  the  benefits  which 
have  come  to  our  race  from  the  pursuit  and  discovery  and  use  of 
the  keys  of  physical  science,  have  been  bestowed  upon  us  through 
these  suggestions  of  geometrical  thoughts  in  the  outward  creation. 

But  in  the  pursuit  of  mathematical  knowledge  men  began,  at  an 


SYMMETRY  IN   SPACE.  5 

early  age,  to  invent  and  investigate  a  priori  laws,  laws  of  which 
they  had  not  received  any  suggestion  from  nature.  And  the 
intellectual  origin  of  the  forms  of  nature  was  made  still  more  mani- 
fest when  these  a  priori  laws,  of  man's  invention,  Avere,  in  many 
cases,  afterwards  discovered  to  have  been  truly  embodied  in  the 
universe  from  the  beginning;  as,  for  example,  Plato's  conic  sec- 
tions in  the  forms  and  orbits  of  the  heavenly  bodies,  and  Euclid's 
division  in  extreme  and  mean  ratio. 

This  division  in  the  extreme  and  mean  ratio  was  invented  by 
the  early  geometers,  without  any  known  suggestion.  It  is  evi- 
dent that  this  division  might  be  illustrated  in  a  great  variety  of 
ways.  A  whole  must  be  divided  into  two  parts,  such  that  the 
first  shall  bear  the  same  relation  to  the  second  that  the  second 
does  to  the  whole.  No  matter  what  the  whole  is,  a  division  of  it 
approximately  in  this  manner  would  be  an  expression  of  the  idea 
of  extreme  and  mean  ratio.  If  the  whole  were  a  quantity  (dis- 
tance, angle,  surface,  volume,  value,  time,  velocity,  &c.),  and  the 
relation  were  that  of  magnitude,  the  whole  would  be  to  the  smaller 
part  as  unity  is  to  half  the  difference  between  three  and  the  square 
root  of  five.  If,  on  the  other  hand,  the  whole  were  a  work  of  art 
of  any  kind,  or  a  system  of  thought,  the  relation  would  not  be 
one  of  mere  magnitude ;  and  the  division  would  be  a  work  of  more 
ingenuity.  But,  whatever  the  whole,  or  the  relation,  the  proper 
division  would  be  an  expression  of  the  idea. 

Now  we  have,  in  nature,  at  least  three  embodiments  of  the  law 
of  extreme  and  mean  ratio,  two  of  which  are  very  striking.  The 
botanists  find  that  two  successive  leaves,  counting  upward  on  the 
stem,  stand  at  an  angle  with  each  other  that  is  either  one-half, 
one-third,  two-fifths,  three-eighths  of  the  whole  circle ;  or  some 
higher  approximation  to  this  peculiar  proportion.  The  seed  ves- 


6  GEOMETRY   AND    FAITH. 

sels  and  buds  on  a  spike  of  broad-leaved  plantain  afford  one  of  the 
most  instructive  examples.  They  are  usually  set  on  a  high  ap- 
proximation, so  that  the  order  is  not  apparent.  Take  a  piece  of 
the  spike,  an  inch  or  so  in  length,  between  your  hands,  and  gently 
twisting  reduces  it  to  three  ;  while  a  slight  twist  in  the  opposite 
direction  brings  out  five  rows,  which  a  harder  twist  reduces  to 
two. 

The  efficient  cause  of  this  arrangement  we  do  not  know.  It  has 
been  ingeniously  suggested  that  it  might  be  produced  by  a  simple 
law  of  the  genesis  of  cells.  Let  us  suppose  that  each  cell  emits 
a  new  cell  at  regularly  recurring  intervals  of  time,  and  that  the 
new  cell  begins  to  generate  cells  at  the  expiration  of  two  intervals 
after  its  birth.  A  cell  developing  on  a  plane,  under  this  law, 
would  produce  its  cells  in  the  phyllotactic  order  of  the  leaves,  in 
the  terminal  rosette  of  a  plant.  But  it  is  difficult  to  see  how  this 
hypothesis  can  be  made  to  include  and  explain  the  whole  phe- 
nomena of  the  arrangement. 

The  final  causes,  although  the  devout  mind  always  recognizes 
the  impossibility  of  man's  attaining  a  certainty  concerning  all  the 
final  causes  of  the  phenomenon,  are  more  obvious.  It  has  been 
shown  that  this  division  of  the  circle  insures  in  the  only  perfect 
way  to  each  leaf  its  chance  at  zenith  light,  its  best  chance  at  air ; 
in  short,  that  this  phyllotactic  law  distributes  the  leaves  most 
evenly  about  the  stem. 

In  the  solar  system,  if  we  divide  the  periodic  time  of  each 
planet  by  that  of  the  planet  next  farthest  from  the  sun,  we  shall 
have,  beginning  with  the  quotient  of  Uranus'  year  divided  by 
that  of  Neptune  and  ending  with  the  quotient  of  Mercury's  year 
divided  by  that  of  Venus,  a  series  of  fractions  agreeing  very 
closely  with  the  approximations  of  the"  phyllotactic  law.  The 


SYMMETRY  IN   SPACE.  7 

problem  was  similar.  The  planets  would  not  have  remained  in 
proper  subjection  to  the  sun  had  they  been  allowed  to  group 
themselves  too  frequently  in  one  rebellious  line,  hanging  upon 
the  golden  chain  of  his  attraction,  dragging  him  and  themselves 
from  their  proper  orbits.  They  must  be  kept  evenly  distributed 
about  the  sun;  and  since  they  are  moving,  the  times  of  their 
revolution,  their  angular  velocities,  must  be  divided  by  the  same 
law  as  that  which  divides  the  stationary  angles  of  the  leaves. 

We  have  then  in  the  plants  a  geometrical  or  angular  illustra- 
tion, and  in  the  planets  an  algebraical  or  temporal  illustration,  of 
the  mathematical  idea  of  extreme  and  mean  rati6.  The  infer- 
ence seems  irresistible,  —  these  two  illustrations,  which  cannot  be 
imagined  as  having  any  causal  or  genetic  connection,  owe  their 
intellectual  relation  to  having  sprung  from  One  Mind. 

This  is  a  striking  illustration,  but  the  same  inference  may  be 
drawn  from  every  form  in  nature,  —  planet,  crystal,  plant,  and 
animals.  All  natural  forms  conform  more  or  less  closely  to  geo- 
metrical ideals ;  sufficiently  near  to  suggest  these  ideals  to  men 
fitted  to  receive  the  suggestion  ;  sufficiently  near  to  show  that  the 
whole  of  nature  may,  in  one  sense,  be  regarded  as  a  series  of  draw- 
ings and  models,  by  which  to  teach  the  mathematics  to  students  in 
the  school  of  life. 

The  final  causes  may  never,  however,  be  considered  as  wholly 
known.  The  perfection  of  the  Divine  workmanship  is  shown  in 
the  adaptation  of  each  object  in  nature  to  a  great  variety  of  ends. 
The  geometrical  laws,  on  which  the  world  is  buijt,  are  adapted  to 
all  the  wants  and  all  the  needs  of  every  creature.  Our  human 
needs  are  innumerably  various,  and  nature  finds  means  to  satisfy 
them  all.  Our  intellect  craves  symmetry,  and  through  symmetry 
is  first  led  to  the  perception  of  geometric  law.  But  we  love  the 


8  GEOMETRY   AND    FAITH. 

symmetry  before  we  perceive  the  law.  The  sense  of  beauty  i? 
satisfied,  even  in  externals,  most  perfectly,  and  fills  us  with  most 
pleasure,  in  things  that  the  understanding  fails  to  analyze  and 
define.  Much  has  been  written  concerning  an  analysis  of  the 
beauty  of  outline  ;  one  great  painter  thinking  it  consists  in  flexure, 
others  assigning  it  to  a  spiral,  or  a  helix,  or  an  ellipse ;  while 
Darwin  refers  it  to  early  association,  while  yet  a  suckling,  with 
the  form  of  the  mother's  breast.  I  venture  with  diffidence  to 
give  my  own  opinion,  that  the  perception  of  beauty  in  outline  is 
the  unconscious  perception  of  geometric  law, — just  as  the  per- 
ception of  harmony  has  been  demonstrated  to  be  the  unconscious 
perception  of  arithmetical  ratios  in  time,  or  algebraic  law.  The 
beauty  of  outline,  I  would  say  of  external  form,  independently  of 
expression,  is  in  proportion  to  the  simplicity  of  the  geometric  law, 
and  to  the  variety  of  the  outline  which  embodies  it.  Nor  is  it 
essential  to  the  highest  enjoyment  of  beauty  that  the  conformity 
to  geometric  ideals  should  be  perfect,  any  more  than  it  is  essential 
to  the  highest  music  to  have  the  harmony  perfect.  On  the  con- 
trary, the  higher  degrees  of  beauty  are  apt  to  be  found  in  forms 
that  suggest,  rather  than  embody,  the  ideal ;  and  especially  in 
figures  potentially,  but  not  actually,  symmetrical.  The  monotony, 
which  might  result  from  unbroken  regularity  of  form,  is  avoided, 
and  a  new  grace  is  given,  for  example,  to  the  higher  animals,  by 
their  temporary  disguise  of  symmetry,  in  their  varied  positions  and 
movements.  In  the  sea  shells,  the  same  end  is  attained  by  the 
spiral  form,  which  so  many  of  them  take  ;  in  which  there  is  not  an 
actual  symmetry,  but  only  a  law  of  symmetry,  the  perfect  develop- 
ment of  which  would  require  an  infinite  number  of  convolutions. 

In  the  forms  of  vegetative  life,  there  is  the   widest  departure 
from  actual  symmetry,  and  yet  a  constant  suggestion  of  its  laws. 


SYMMETRY  IN   SPACE.  9 

The  phyllotactic  law  secures  to  the  tree  a  general  regularity,  and 
equal  growth  upon  every  side  ;  and  yet,  by  complication  of  detail, 
combined  with  occasional  failure  or  destruction  of  buds,  secures 
an  endless  variety  of  graceful  forms,  in  each  species.  May  we 
not  then  name  beauty  as  another  final  cause,  another  end  secured 
by  the  adoption  of  the  division  in  extreme  and  mean  ratio  ?  The 
approximations  are  beautiful  to  us,  and  the  pleasure  given  to  us 
was  foreseen  when  the  law  was  adopted.  May  it  not  also  have 
been  felt ;  and  may  not  the  forms  of  flowers  be  but  approximations 
toward  the  expression  of  an  infinite  beauty,  hidden,  from  all  finite 
sense,  in  the  incommensurable  ratio  of  that  surd  ?  That  the  ex- 
ternal symmetry  of  animals  may  have  beauty  as  its  final  cause,  is 
rendered  probable  from  the  lack  of  symmetry  in  the  viscera,  which 
are  hidden  from  sight. 

Whatever  be  our  speculations  upon  such  points,  this  at  least  is 
manifest,  that  the  sense  and  the  presence  of  beauty  are  kindly 
adapted  to  each  other  hi  the  world.  Even  shapeless  matter  de- 
clares its  Creator's  power;  the  perfect  symmetry  of  crystalline 
forms,  the  potential  symmetry  of  all  the  organic  worlds,  show  forth 
His  wisdom  and  His  love. 


II. 


SYMMETRY  IN  TIME. 

TIME  has  but  one  dimension,  and  is  divisible  only  into  before 
and  after.  In  the  zero  of  now,  the  future  is  becoming  the  past ; 
and  this  suggests  the  division  of  the  future,  and  of  the  past,  by  the 
insertion  of  imagined  presents,  zero  boundaries,  dividing  time  into 
periods ;  these  imagined  future  nows  becoming  actual,  as  we  suc- 
cessively reach  them,  and  those  past  having  been  actual,  as  we 
passed  them.  As  time  flows  only  forward,  the  imagination  runs 
backward  into  the  past  with  the  greatest  difficulty ;  indeed  I  am 
not  certain  whether  it  is  possible  for  the  imagination  to  run  back ; 
when  I  attempt  to  do  so,  I  find  that  I  leap  backward,  by  longer  or 
shorter  leaps,  but  never  run  continuously  in  imagination  through 
time,  except  forward,  from  the  moment  to  which  I  have  leaped. 

By  symmetry  in  time,  therefore,  we  do  not  mean  a  similar  ar- 
rangement of  intervals  before  and  after  a  certain  moment.  This 
has  occasionally  been  attempted  in  per  rede  et  retro  chanting  ;  but 
it  is  a  transference  of  geometrical  symmetry  to  time,  where  it  is 
out  of  place,  and  tasteless.  Symmetry  in  time  is  the  arrange- 
ment of  two  or  more  similar  series  of  intervals,  to  follow  the  same, 
or  successive  movements.  When  the  set  of  similar  intervals  fol- 
lows the  same  moment,  it  constitutes  keeping  of  time  ;  when  suc- 
cessive moments,  rhythm;  unless  the  intervals  are  very  short, 
when  rhythm  becomes  tone,  or  color,  and  keeping  time  becomes 
harmony. 

(10) 


SYMMETRY   IN   TIME.  11 

The  passion  for  harmony  and  rhythm  is  an  essential  element  in 
human  nature.  It  is  a  passion  which  varies  greatly  in  intensity  in 
different  persons,  but  it  is  never  wholly  absent.  Savage  nations 
ha/e  some  rudiments  of  music  and  of  song.  The  naked  Fuegian, 
when  the  stormy  winds  of  his  inhospitable  straits  pause  for  a  while 
in  their  wild  uproar,  chants  his  songs  or  hymns  in  a  rude  measure 
and  melody.  The  dullest  ear  for  harmony  has  an  ear  for  rhythm 
sufficient  to  perceive  the  difference  between  prose  and  verse.  It 
is  when  the  intervals  of  time  in  rhythm  become  so  short,  as  to  be 
'separately  imperceptible,  that  the  rhythm  is  called  simply  tone  ; 
and  harmony  is  the  simultaneous  movement  of  tones.  Man  finds 
pleasure  in  all  forms  of  symmetry  in  time,  whether  the  parts  are 
perceptible,  or  imperceptibly  short ;  and  the  world  has  been  made 
in  exquisite  adaptation  to  this  taste  of  man. 

Space  has  three  dimensions,  time  but  one.  Yet,  in  some  re- 
spects, time  is  richer  in  its  contents  for  man,  than  is  space.  The 
beauty  of  forms  in  space,  is  almost  equalled  by  the  beauty  of 
color,  and  color  arises  simply  from  symmetry  of  times  ;  it  is  a  kind 
of  tone.  Color,  indeed,  is  more  expressive,  more  directly  produc- 
tive of  pleasure  to  the  eye,  than  form.  The  latter  appeals  more 
to  the  intellect,  and  is  more  directly  expressive  of  intellectual 
ideas ;  the  former  appeals  more  to  the  heart,  and  gives  a  sweeter 
pledge  of  the  Divine  Love.  But  beside  color,  symmetry  in  time 
gives  us  music  in  all  its  vast  variety  of  forms  and  expressions.  In 
music  there  is  a  beauty  as  distinctly  intellectual  as  that  of  geo- 
metrical figures,  and  a  power  of  expression  which  geometric  form 
attains,  scarcely  even  in  the  human  figure  and  face.  Nor  can  we 
omit  to  mention  heat,  which  although  not  giving  direct  pleasure 
to  the  mind,  and  the  heart,  as  beauty,  color,  and  music  do,  is 
still  essential  to  the  life  of  the  body  and  to  its  comfort  Many 


12  GEOMETRY   AND   FAITH. 

chemical  changes  are  also  produced  by  minute  symmetric  mo- 
tions. 

A  minute  symmetrical  division  in  space  produces  no  sensation 
for  us,  except  as  it  may  lead  to  symmetrical  motion.  Thus  the 
minute  symmetry  of  the  particles  of  a  solid  in  a  clear  liquid  solu- 
tion, is  revealed  to  us  only  through  the  motion  of  light,  and  the 
changes  which  that  motion  experiences  in  going  through  the  solu- 
tion. In  like  manner  the  temporal  symmetry  of  motion  in  the  ray 
of  light  coming  from  the  stars,  brings  us  more  information  than  its 
mere  property  of  making  visible  can  give ;  for,  on  cross-examina- 
tion by  the  spectroscope,  it  confesses  the  chemical  secrets,  and  the 
degree  of  heat  in  the  star  at  the  time  of  its  leaving,  ages  ago ; 
and  also  the  direction  of  the  star's  motion. 

Man's  organization,  and  his  surroundings,  are  adapted  to  his 
love  of  music.  His  voice  is  capable  of  being  regulated  to  musical 
tones  of  various  pitch.  The  metals  are  sufficiently  elastic  to 
render  them  sonorous  ;  and,  as  the  foundation  of  all,  the  air  itself, 
by  its  elasticity,  becomes  the  vehicle  of  sound  and  the  instrument 
of  music. 

Two  gases,  intermingled,  remain  to  a  certain  extent  indepen- 
dent of  each  other ;  and,  inasmuch  as  sound  travels  in  each  gas 
with  a  velocity  proportioned  to  the  density  and  elasticity  of  that 
gas,  there  will  be,  from  a  single  source  of  sound,  two  sounds  prop- 
agated in  the  mixture,  with  different  velocities,  interfering  with 
each  other,  and  destroying  the  pure  tone  of  a  musical  sound. 
Now  the  atmosphere  is  a  mixture  of  heavy  oxygen,  with  lighter 
nitrogen.  The  elasticities  are,  however,  so  nearly  adjusted  to  the 
densities,  that  sounds  travel  in  either  gas  with  nearly  the  same 
velocity,  so  that  the  air  sounds  in  an  organ  pipe,  as  if  -one  gas. 
Had  sound  travelled  in  these  two  gases  at  rates  differing  as  much, 


SYMMETRY    IN    TIME.  13 

as  the  rate  in  them  differs  from  that  in  most  of  the  gases  known  to 
us,  the  use  of  wind  instruments  of  music  would  have  been  impos- 
sible ;  probably  all  music,  even  the  tones  of  the  human  voice, 
would,  in  that  case,  have  been  discordant  to  an  ear  at  any  consid- 
erable distance  from  the  source  of  sound.  With  the  intense  and 
elevating  character  of  the  pleasure  derived,  first  from  the  tones  of 
human  speech,  from  the  melody  of  birds,  and  other  natural  music, 
and  secondly  from  the  art  of  music,  in  our  minds,  we  cannot  but 
be  grateful  for  this  adaptation  of  the  mingled  atmosphere  to  the 
needs  of  man,  in  his  higher  nature. 

All  undulatory  motion  produces  a  symmetrical  division  of  time. 
The  beauty  of  color,  like  that  of  tone,  arises  from  an  implicit  per- 
ception of  rhythm.  The  harmony  of  tints  in  the  landscape, 
like  that  of  the  sounds  in  a  strain  of  music,  arises  from  the  har- 
mony of  times  in  which  the  vibrations  of  the  mediums  occur.  The 
pleasure,  in  either  case,  arises  from  an  implicit,  or  unconscious, 
perception  of  keeping  time.  Heat  also  has  its  colors,  or  tones,  as 
is  known  to  all  who  have  noticed  that  the  sun's  heat  passes  freely 
through  glass ;  which  is  impervious  to  the  heat  of  a  fire.  What 
other  advantages  to  man  may  hereafter  be  discovered,  in  this 
coloration  of  heat,  time  alone  can  show ;  but  when  we  consider  to 
what  an  extent,  through  the  providence  of  God,  glass  is  employed, 
it  seems  not  irreverent  to  own  our  gratitude  to  Him  that  this  sub- 
stance reflects  back  the  warmth  of  our  apartments,  and  keeps  it 
within,  but  allows  the  heat  of  the  sun  to  pass  through  from  with- 
out. 

Besides  these  hidden  proofs  of  creative  foresight,  and  benefi- 
cence, in  the  concealed,  or  minute  symmetry  of  time,  we  shall 
find  open  and  abundant  proofs  in  manifest  rhythmic  movements. 
In  the  play  of  alternating  muscles,  we  perceive  an  adaptation  of  the 


14  GEOMETRY   AND    FAITH. 

physical  frame  to  the  intellectual  taste.  In  walking,  for  example, 
there  are  few  persons  who  do  not  feel  the  increase  of  pleasure  and 
of  power  gained  by  keeping  step.  A  single  drum-tap,  regulating 
the  tramp  of  a  large  body  of  men,  has  sometimes  an  effect  almost 
equal  to  that  of  music.  The  rhythm  of  verse,  and  of  music,  de- 
lights many  who  are  comparatively  insensible  to  both  melody  and 
harmony  properly  so  called.  Who  that  reflects  upon  the  genius  of 
Bach,  of  Handel,  of  Haydn,  and  Beethoven,  and  considers  the 
effect  which  music,  such  as  theirs,  has  upon  the  world,  can  doubt 
the  kindness  of  that  superintending  Power,  who  kindled  the  fire 
in  their  hearts,  and  through  that  in  ours ;  who  also  adapted  the 
air,  and  the  various  materials,  for  man,  by  which  he  pours  out  his 
musical  conceptions  ?  Who  that  reflects  upon  the  genius  of  a 
Homer,  and  a  Shakespeare,  and  remembers  to  how  many  millions 
their  verse  has  given  delight  and  instruction,  can  doubt  that  the 
same  Beneficence  gave  the  poet  his  power,  and  men  the  heart  to 
be  touched  with  poetry  ? 


III. 

NUMBER. 

NUMBER  Is  not,  like  space  and  time,  matter  and  spirit,  an  inte- 
gral part  of  the  Universe,  nor  is  it  a  necessary  attribute  of  either 
of  these.  Space  and  time  are  without  parts  or  limits,  and  are,  in 
themselves,  so  diverse  that  they  would  not  suggest  even  the  idea 
of  duality.  Thus  also  amorphous  matter  suggests  no  number. 
Number  is  an  impress  of  thought,  it  is  a  pure  creation  of  Spirit ; 
and  its  constant  suggestion  in  the  forms  and  periods  of  nature,  is 
a  clear  demonstration  that  nature  is  the  work  of  an  Intellect  which 
controls  both  space  and  time  in  thought.  The  human  intellect 
early  learns  number  from  the  text-book  of  outward  nature ;  and 
delights  in  tracing,  further  than  nature  goes,  the  laws  of  number. 

From  the  great  usefulness  of  this  earliest  abstract  science,  and 
from  the  fascination  of  its  pursuit,  arithmetic  has,  in  modern 
schools,  been  allowed  to  usurp  the  place  of  geometry ;  and  the 
pupil  has  been  taught  to  reason  upon  abstract  numbers  before  he 
has  learned  to  conceive  clearly  imaginary  forms.  From  the  same 
fascinating  power,  number  has  sometimes,  in  the  minds  of  great 
men,  like  Pythagoras,  been  allowed  to  occupy  a  disproportionate 
share  of  attention  ;  as  though  number  included  all  proportion  and 
beauty.  Even  the  Hebrews,  with  all  their  clearer  light  of  truth, 
appear  to  attach  a  mysterious  power  to  number. 

There  is  a  power  in  number.  When  our  human  thought  at- 
tempts the  survey  of  space  and  time,  and  would  subdue  these 


16  GEOMETRY   AND    FAITH. 

realms  to  obedience  under  our  intellect,  we  find  ourselves  com- 
pelled, before  we  can  attain  any  precision  in  our  forms,  to  intro- 
duce number.  The  reason  can  deal,  to  some  extent,  with  con- 
tinuous quantity,  moving  under  continuous  law,  and  not  in  the 
proportion  of  numbers.  But  the  imagination  cannot  take  a  step 
with  any  clearness,  much  less  can  the  hand  build  with  any  satis- 
faction, without  referring  quantities  to  a  unit  of  quantity,  to  which 
the  ratio  shall  be  that  of  two  numbers  to "  each  other.  And  of 
course  our  finite  intellects  handle  with  most  ease  the  smaller  num- 
bers ;  so  that  these  become  to  us  the  most  important ;  and  there  is 
not  a  number  under  ten  which  has  not  some  strong  associations 
with  it  in  the  human  mind,  which  give  it  a  kind  of  sanctity. 
These  mystic  charms  cluster  especially  around  the  odd  numbers 
three,  five,  seven,  and  nine ;  which  seem  to  have  an  individuality ; 
the  first-named  three  being  primes,  while  four  is  but  two  twos ; 
and  six,  two  threes ;  and  these  charms  were  felt  in  the  earliest 
ages  of  human  history. 

But  nature  also  loves  these  numbers  ;  and  they  are  illustrated, 
even  to  the  untaught  mind,  by  many  phenomena ;  organic  beings 
possess  a  unity,  which  is  absolute ;  the  sexes,  of  both  plants  and 
animals,  give  us  duality ;  the  powers  within,  and  those  above,  sug- 
gest the  threefold  division ;  the  points  of  the  compass,  the  limbs  of 
mammals,  give  us  the  number  four ;  the  fingers  of  the  hand,  five, 
and  so  on.  And  the  increasing  knowledge  of  the  physical  world, 
in  our  nineteenth  century,  brings  us  increasing  proof  that  God, 
who  planned  heaven  and  earth,  was  acquainted  with  numbers ; 
made  all  things  in  number,  weight,  and  measure ;  and  adopted 
the  smaller  numbers,  either  out  of  preference  for  them,  or  in 
condescension  to  the  minds  of  his  children,  whom  he  has  placed 
here  for  their  preparatory  education. 


NUMBER.  17 

Chemistry  is  a  science  of  this  century,  and  it  teaches  us  that, 
from  the  beginning,  the  numbers  two  and  three  have  been  domi- 
nant powers  in  the  Universe.  Simple  unites  with  simple  to  form 
a  couple,  a  compound.  This  couple  rarely  takes  a  third  element 
to  form  a  triplet.  The  'couplets  and  triplets  unite  again  in  com- 
pound couplets,  and  thus  the  innumerable  variety  of  substances 
is  built  up  under  the  simplest  possible  combinations  of  number. 
Follow  these  substances  through  all  their  various  modes  of  mo- 
tion and  action ;  in  their  weights,  in  their  attractions,  their  gas- 
eous condition,  their  volume,  their  specific  heat,  their  color  at  a 
high  temperature ;  and  they  are  found  still  to  be  bound  together 
by  simple  laws  of  arithmetical  proportion. 

Consider  also  the  law  of  extreme  and  mean  ratio,  as  exhibited  in 
the  leaves  of  plants.  In  itself  the  law  transcends  the  power  of 
number,  and  had  the  plants  fulfilled  it  with  absolute  accuracy,  it 
might  have  been,  even  yet,  hidden  from  the  mathematician's  eye. 
But  the  plants  give  it  to  us  only  by  approximations ;  approxima- 
tions which  demonstrate  that  the  exact  law  was  known  to  the 
Builder  of  the  plant,  and  is  by  him  revealed  to  the  mathemati- 
cian ;  but  which  give  to  the  unlearned  the  simpler  conception  of 
the  first  four  prime  numbers ;  in  the  beautiful  varieties  of  leaves 
opposite,  and  leaves  ternate,  five  pointed  and  seven  pointed  stars. 

The  laws  of  musical  harmony  are  especially  to  be  noted.  When 
the  waves  of  the  air  are  perceived  only  as  continuous  musical 
tones,  and  the  individual  vibrations  are  not  at  all  recognized,  why 
should  the  ratio  of  four  to  five  give  us  pleasure,  and  that  of  eight 
to  eleven  give  us  none  ?  What  process  of  education  in  our  ances- 
try, what  association  of  ideas,  renders  the  effect  of  the  one  com- 
bination harmonious,  of  the  other  discordant?  Any  attempt  to 
explain  it  will  but  strengthen  the  conclusion,  that  to  the  Builder 


18  GEOMETRY   AND    FAITH. 

of  the  ear  the  laws  of  number  were  known,  and  that  the  ear  was 
constructed  with  reference  to  them. 

The  harmonies  of  light  and  heat  are  not  sufficiently  well  under- 
stood to  make  the  argument  here  so  apparent.  Yet  there  is, 
doubtless,  in  these  departments  also,  an  adaptation  of  the  human 
sense  to  the  perception  of  effects  arising  from  simple  numerical 
proportions  in  the  frequency  of  vibrations.  In  the  matter  of 
geometric  form,  while  the  value  of  proportion  has  been  felt  by  all 
artists,  and  all  architects,  the  value  of  numbers  in  the  proportions 
has  not  been  universally  conceded,  nor  its  place  assigned.  Yet 
I  have  by  experiments,  upon  unprejudiced  persons  of  good  taste, 
strengthened  greatly  my  inclination  to  accept  Hay's  law,  —  that 
angles,  real  or  potential,  are  the  essential  elements  of  geometric 
beauty ;  and  are  beautiful  in  proportion  to  the  numerical  sim- 
plicity of  their  ratio  to  the  right  angle. 

With  these  manifest  indications  that  the  divine  thought,  the 
ideals  of  the  creation,  include  number  as  an  essential  element,  we 
may  well  understand  the  enthusiasm  of  early  thinkers  over  the 
properties  of  the  smaller  numbers.  The  sacredness  of  the  num- 
ber three  has  been  made  especially  prominent  in  Christendom. 
The  four  elements  of  the  ancients,  and  Erigena's  fourfold  division 
of  nature,  show  the  power  of  the  points  of  the  compass  to  impress 
their  number  on  the  human  mind.  The  five  digits  of  the  hand, 
and  the  prevalence  of  fivefold  divisions  in  the  floral  kingdom,  give 
us  the  five-pointed  star  with  its  symbolism  ;  point  up,  for  manhood 
and  virtue;  point  down,  for  beastliness  and  sin.  The  lily  tribe 
gives  us  the  six-pointed  star ;  and  six,  a  perfect  number,  in  which 
the  sum  of  the  factors  equals  the  product,  is  fitting  as  a  symbol  of 
the  descent  of  the  divine  into  the  human  trinity,  the  indwelling  of 
God  in  man ;  the  Perfect  perfecting  his  child.  The  seven  notes 


NUMBER.  19 

of  the  diatonic  scale,  the  seven  distinct  colors,  and  other  natural 
examples,  fall  in  with  the  seven  days  of  the  week,  the  quartering 
of  the  moon's  period.  'Jew  and  Gentile  alike  have  hallowed  the 
number  seven,  and  no  other  number  occurs  so  frequently  with 
sacred  associations  in  Jewish  and  Christian  literature.  Higher 
primes  than  seven  do  not  enter  much  into  our  human  thought,  nor 
appear  to  be  embodied  distinctly  in  any  part  of  creation  known  to 
us.  The  weeks  in  the  year  are  four  times  thirteen  ;  that  is,  there 
are  about  thirteen  moons  in  the  year ;  the  only  example  I  re- 
member of  a  prime  number  above  seven  prominently  suggested 
by  nature.  The  nine  muses,  the  ten  numerals,  the  twelve  months, 
and  twelve  apostles  are  numbers  not  prime. 

Music,  painting,  the  coloring  of  nature  and  art ;  architecture, 
sculpture,  drawing,  the  beauty  of  proportion  and  form ;  how  large 
a  portion  of  our  earthly  pleasure  and  spiritual  culture  depends 
on  these  ;  and  these  draw  their  charm  in  some  mysterious  way 
from  the  numbers  two,  three,  five,  seven.  The  number  of  prime 
numbers  is  unlimited  ;  and  since  the  first  four  give  us,  in  the  har- 
mony of  tones  and  colors,  and  in  the  proportions  of  form,  such 
varied  sources  of  high  pleasure,  such  potent  modes  of  spiritual  ex- 
pression, we  may  reverently  hope,  that  in  the  immortal  life,  the 
same  Beneficent  Power  which  makes  two,  three,  five,  and  seven, 
thus  minister  to  our  joys  below,  will  open  to  us  more  of  the  infinite 
treasures  which  lie  hidden  in  the  boundless  fields  beyond. 


IV. 

THE  CALCULUS. 

SPACE  and  time  are  so  entirely  diverse  in  their  nature,  that 
there  is  no  connection  or  relation  between  them ;  except  through 
the  mind,  as  percipient  of  both ;  or  through  will,  manifesting  itself 
in  motion.  In  contemplating  space  we  see  it  as  external  to  the 
mind  ;  our  consciousness  does  not  sharply  locate  its  own  where- 
abouts ;  we  fancy  ourselves  near  the  Eyegate  or  Eargate  of  the 
town  of  Mansoul;  but  cannot  say  precisely  where  our  council 
chamber  may  be  situated.  Not  so  with  time,  our  consciousness  is 
sharply  defined  ;  we  are  neither  in  the  past  nor  in  the  future  ;  our 
conscious  moment  is  the  now,  without  duration.  Hence  we  can 
more  readily  imagine  ourselves  freed  from  limitations  of  space  than 
from  those  of  time.  We  can  imagine  to  ourselves  time  in  the  flow 
of  our  own  thoughts  ;  the  thought  of  space  necessarily  takes  us 
out  of  ourselves.  But  when  we  go.  out  of  ourselves  and  contem- 
plate space,  we  carry  time  with  us  in  the  very  action  of  our 
thought.  In  all  closer  contemplation  of  outlines,  the  attention  is 
transferred  successively  to  different  points  of  the  figure,  and  time 
is  occupied  by  that  transfer.  Thus  we  come  naturally,  and  almost 
inevitably,  to  regard  the  line  as  the  path  of  a  moving  point,  the 
surface  as  generated  by  a  moving  line. 

Thus  space  and  time,  though  heterogeneous,  are  united  into  one 
science  of  mathematics  by  human  thought ;  and  the  laws  of  alge- 
bra, or  time,  are  applied  to  geometry,  or  space.  By  this  simple 

(20) 


THE    CALCULUS.  21 

device,  into  which  Descartes  and  Newton  were  led  by  nature's  own 
guidance,  the  human  mind  has  extended  almost  indefinitely  its 
geometrical  acquisitions  ;  it  was  by  carrying,  as  it  were,  its  native 
element  of  time  with  it  into  the  domain  of  space  that  it  has  con- 
quered so  vast  a  field. 

When  we  remember  how  intense  the  delight  which  man  feels  in 
the  discovery  of  mathematical  truths ;  how  many  of  the  noblest 
thinkers  of  the  race  have  owed  their  finest  discipline  to  this  pur- 
suit ;  how  rich  the  harvest  of  practical  benefits  which  have  flowed 
from  the  application  of  mathematics  to  the  arts  and  sciences  ;  how 
magical  their  effect  has  been  in  banishing  superstition,  and  elevat- 
ing the  general  tone  of  human  thought  and  human  endeavor,  we 
may  surely  own,  with  gratitude,  the  marks  of  divine  wisdom  and 
love,  in  this  gift  to  man,  of  the  power  to  penetrate  space,  and  ap- 
ply to  it  the  laws  of  time.  It  is  a  peculiar  gift,  not  a  necessary 
accompaniment  of  intellect,  for  sometimes  the  brightest  intellects 
possess  it  in  only  a  very  feeble  degree.  Thankfully,  therefore,  do 
we  acknowledge  the  presence  of  an  Infinite  Spirit,  giving  good 
gifts  to  man  in  the  inspiration  of  a  Leibnitz  and  a  Lagrange,  as 
well  as  of  a  Handel  and  a  Shakespeare. 

The  main  source  of  this  power  given  by  algebra  to  the  geometer, 
is  the  comprehensiveness  of  the  language  put  into  his  hands.  The 
introduction  of  general  and  abstract  terms  is  always  a  means  of 
enlarging  the  grasp  of  thought,  and  increasing  the  clearness  of 
reasoning.  Space  has  its  three  dimensions,  its  elements  of  magni- 
tude and  direction  ;  and  although,  in  one  aspect,  the  simplest  of 
all  possible  objects  of  thought,  may  yet,  for  purposes  of  reasoning 
concerning  it,  be  advantageously  reduced,  by  algebraical  language, 
to  the  one  term  of  quantity,  capable  only  of  flowing  in  one  direc- 
tion, and  being  considered  as  greater  or  less  than  a  given  magni- 


22  GEOMETRY  AND   FAITH. 

tude.  But  the  generality  thus  introduced  is  made  vastly  more 
general  by  using  symbols  which  shall  combine,  in  one  letter,  vari- 
ous forms  and  relations  in  space,  defined  according  to  judiciously 
selected  and  easily  interpreted  laws.  Thus,  for  example,  all  possi- 
ble triangles,  plane  and  spherical,  and  all  their  properties  are  im- 
plied in  the  single  equation,  r  =  pq ;  and  a  similar  condensation 
of  meaning  is  attained  in  mechanical  science.  Another  source  of 
the  peculiar  power  of  the  calculus  arises  from  the  plasticity  which 
it. gives  to  infinitely  rigid  space.  In  experimenting  upon  a  rec- 
tangular beam,  cut  from  a  round  piece  of  timber,  we  can  readily 
determine  its  strength  when  set  edgewise ;  but  cannot  tell  what 
the  strength  would  have  been  had  the  sides  been  in  different  pro- 
portions. The  rectangular  parallelepiped  inscribed  in  a  cylinder 
is  as  absolutely  fixed  in  its  dimensions  as  the  hewn  timber,  but  by 
expressing  those  dimensions  in  language  borrowed  from  the  science 
of  time,  we  can  imagine  them  changing  in  their  proportions,  and 
the  strength  changing  with  them.  Thus  we  can  determine  the 
precise  proportion  they  must  bear  in  order  to  give  the  strongest 
possible  rectangular  beam  that  could  be  cut  from  a  round  log. 
This  illustrates,  by  a  simple  example,  the  power  given  to  geome- 
try by  Newton's  conception  of  fluxions,  his  introduction  of  the 
idea  of  velocity  into  the  consideration  of  form. 

The  appearance  of  the  same  algebraic  law  in  the  creation,  under 
the  two  forms  ~>f  time  and  space,  has  already  been  alluded  to  as 
proof  of  unity  of  design  ;  the  angles  of  leaves  and  the  angular  ve- 
locity of  planets  being  expressed  by  the  same  series  of  fractions. 
Other  examples  confirm  the  sublime  induction.  The  elasticities  of 
gases,  strings,  and  rods  are  so  fundamentally  different  in  kind  that 
we  see  no  connection  between  them.  The  elastic  force  of  the 
stretched  string  we  need  not  determine ;  that  of  the  rod,  and  that 


THE    CALCULUS.  23 

of  the  gas,  can  be  determined  only  by  experiment,  aud  when  deter- 
mined they  have  no  very  apparent  connection  or  relation  with  each 
other.  Nevertheless,  each  of  the  three  has  a  peculiar  relation  to 
the  force  of  gravity ;  of  which  it  is,  nevertheless,  entirely  indepen- 
dent. The  velocity  of  a  sound  traveling  in  the  air,  near  the  earth, 
would  be,  were  no  heat  developed  in  the  action,  equal  to  the  velocity 
acquired  by  a  body  falling  from  a  height  equal  to  that  which  the 
atmosphere  would  have  could  it  be  all  compressed  to  the  density 
of  that  near  the  earth's  surface.  The  velocity  of  a  wave  traveling 
on  a  string  is  equal  to  that  which  would  be  acquired  by  a  body  fall- 
ing from  a  height  measured  by  the  length  of  the  same  cord  equal 
in  weight  to  the  tension  of  the  string.  And  if  we  take  a  very 
fine  glass  thread  by  its  two  ends,  the  infinitely  varied  and  beauti- 
ful forms  which  it  can  be  made  to  assume,  of  waves  and  folds  and 
kinks  and  loops,  the  figure  eight  and  the  circle,  are  all  expressed 
in  mathematical  language  by  the  same  forms  as  those  which  ex- 
press the  motions  of  an  ordinary  pendulum,  under  the  forces  of 
gravity.  The  genetic  connection,  between  these  forms  and  these 
motions,  we  do  not  see,  any  more  than  that  between  the  times  of 
the  planets  and  the  angles  of  the  leaves,  but  the  intellectual  con- 
nection we  detect,  and  it  leads  us  to  recognize  with  reverential 
awe  the  presence  of  Intellect  in  the  disposition  of  the  particleg 
of  both  gaseous  and  solid  bodies. 


V. 

APPLIED   MATHEMATICS. 

PERFECT  symmetry  belongs  only  to  the  ideal,  not  to  the  actual. 
The  algebraic  conditions  are  exactly  fulfilled  by  points  of  space, 
in  an  invisible  and  eternal  reality ;  to  this  real  form,  conforming 
to  the  algebraic  ideal,  the  material  embodiment  makes  at  least  a 
rude  approximation.  The  algebraist  devises  conditions  of  various 
degrees  of  complexity,  delighting  chiefly  in  the  simplest ;  and 
especially  in  those  giving,  with  simplicity  of  conditions,  the 
greatest  variety  of  resulting  forms. 

The  symmetrical  forms  of  nature  suggest  to  man  the  invention 
of  laws  of  symmetry,  at  first  simply  to  explain  nature,  then  to  an- 
ticipate her  work  ;  leading  to  new  examinations  of  that  work. 
Thus  the  great  mathematical  sciences  have  been  alternately  the 
creation  and  the  creators  of  physical  science.  The  physicists 
have  been  prone  to  deny  that  the  mathematics  constitute  a  sci- 
ence ;  they  have  inclined  to  pronounce  them  only  a  key  to  science, 
a  convenient  language  wherein  to  discuss  the  problems  of  matter 
and  motion.  The  mathematicians,  on  the  other  hand,  whom  we 
should  naturally  consider  the  best  judges  of  what  their  own  work 
is,  have  declared  that  geometry  is  the  science  of  space,  algebra 
the  science  of  time,  and  that  these  are  simply  the  first  subjects 
handled  by  the  human  intellect  with  sufficient  freedom,  vigor,  arid 
precision  to  enable  us  to  draw  necessary  conclusions.  As  for 
geometry  and  algebra  being  mere  keys  to  physics,  the  mathema- 

(24) 


APPLIED    MATHEMATICS.  25 

tician  would  sooner  declare  the  whole  visible  creation  a  mere  set 
of  models  and  diagrams  wherewith  to  illustrate  the  laws  of  space 
and  time.  Whichever  of  these  conflicting  views  is  right,  it  is  un- 
questionable that  the  highway  to  the  temple  of  truth  leads  alter- 
nately from  mathematics  to  physics.  Observation  alone  can  lead 
to  nothing,  without  insight,  —  without  that  clearness  of  inward 
vision  which  sees  more  than  the  outward  fact,  sees  the  divine  ideal 
which  the  fact  partially  embodies. 

Now  in  this  sublime  ascent  to  knowledge  the  first  steps  are 
easiest,  and  the  way  to  them  has  been  made  exceedingly  plain 
and  attractive.  "  In  the  beginning  the  Creating  Spirit  embodied, 
in  the  material  universe,  those  laws  and  forms  of  motion  which 
were  best  adapted  to  the  instruction  and  development  of  the 
created  intellect."  The  circle  and  the  ellipse  are  among  the  sim- 
plest of  figures,  defined  by  the  simplest  laws.  Accordingly  the 
Creator  has  strewn  examples  of  the  circular  form  around  us  on 
every  side  ;  and,  by  the  pictured  alphabet  of  the  heavens,  called 
our  attention  to  the  consideration  of  elliptical  orbits.  When,  in  the 
course  of  ages,  some  of  the  comparatively  easy  problems  of  as- 
tronomy had  been  successfully  solved,  problems  of  more  difficulty 
were  gradually  brought  into  view  ;  and  phenomena  which  were 
not  obvious,  not  pictured  alphabet,  but  the  fine  print  of  creation, 
led  men  into  the  hidden  knowledge  of  optics,  electricity,  chem 
istry,  and  other  forms  of  molecular  physics.  The  course  of 
history  and  of  scientific  progress  has  been  precisely  what  it  might 
have  been  had  God  designed  to  educate  men  ;  to  reason  with 
them  and  teach  them  the  sciences ;  for  there  has  been  a  con- 
stant presentation  of  simpler  truths,  whereby  men  have  been 
led  to  the  acknowledgment  of  those  less  obvious  ;  and  this  is 
essentially  reasoning. 


26  GEOMETRY  AND    FAITH. 

Four  centuries  before  the  Christian  era,  the  mathematicians  of 
Greece  were  lured  into  the  study  of  the  conic  sections  ;  and  this 
prepared  the  way  for  the  mathematicians  of  later  ages  to  discuss 
fully  certain  equations  of  the  second  degree.  These  were  suf- 
ficient for  all  the  more  obvious  phenomena  of  astronomy  and  me- 
chanics ;  and  as  the  demand  for  higher  mathematics  has  been 
made  by  physics,  the  supply  has  been  granted.  The  faith,  which 
prompts  the  scientific  investigator  to  his  labor,  he  may  never  have 
expressed  in  words,  but  his  actions  show  us  what  it  is,  —  an  in- 
born, ineradicable  conviction  that  the  outward  universe  is  intelligi- 
ble, and  shall  at  some  day  be  understood.  But  that  day  ever  re- 
cedes, into  the  glorious  future,  as  we  approach  it ;  the  rate  of 
scientific  progress  increases  from  decade  to  decade,  and  yet  the 
new  problems,  and  the  new  instruments  for  their  solution,  in- 
crease more  rapidly.  The  Divine  Intellect  can  never  be  ex- 
hausted by  the  human. 

A  more  detailed  examination  of  the  history  of  the  separate 
sciences  would  only  confirm  our  conclusion,  that,  in  the  selection 
of  laws  under  which  to  subject  the  universe,  God  has  chosen,  for 
those  things  which  would  first  press  themselves  upon  man's  atten- 
tion, those  which  are  most  readily  interpreted  by  man's  intellect ; 
and  employed  more  intricate  laws  for  things  which  would  natur- 
ally escape  man's  notice  until  the  state  of  mathematical  science 
enabled  him  to  take  higher  problems  ;  in  which  we  recognize  evi- 
dence of  that  kindness  and  foresight,  that  care  for  our  education 
and  our  growth  in  knowledge  and  wisdom,  which  is  an  inspiring 
pledge  to  us  that  we  are  indeed  children  of  the  Most  High. 


VI. 


MOTION. 

THE  universe  about  us  is  in  motion.  Nothing  on  which  the  eye 
can  fall,  or  the  existence  of  -which  the  hand  of  science  can  demon- 
strate, is  at  rest.  The  sun  rises  and  sets,  the  moon  waxes  and 
wanes,  the  very  stars  are  in  motion,  to  the  telescopic  eye.  Clouds 
drive  over  the  heavens,  and  billows  roll  over  the  deep;  vapors  rise 
from  the  ocean,  rivers  run  to  the  sea,  and  the  free  winds  play 
around  the  globe.  Plants  are  ever  growing  or  decaying ;  and 
animals  maintain  their  waste,  or  their  waste  consumes  them.  Our 
modern  theories  show  that  the  sensible  properties  in  inanimate  and 
apparently  motionless  matter,  such  as  temperature,  color,  weight, 
are  really  modes  of  motion  in  the  particles  of  matter ;  and  this 
re-echoes  the  sublime  statement  of  the  earliest  seer,  that  the  intro- 
duction of  motion  into  the  universe  was  the  first  act  of  creation. 

For,  upon  a  closer  examination  of  motion,  and  more  accurate 
investigation  of  its  laws,  what  do  we  find  ?  That  the  first  law  of 
motion  is  this :  A  body,  free  from  external  influence,  moves  with 
uniform  velocity  in  a  straight  line  forever.  This  is  the  first  law  of 
motion,  derived  from  the  widest  generalizations,  by  legitimate  in- 
duction from  observations,  on  an  immense  variety  of  motions,  in 
nature,  and  in  the  laboratory.  But  to  what  an  astonishing  result 
does  this  law  lead  us  when  we  apply  it  to  the  case  of  a  body  at 
rest,  the  velocity  of  which  is  nothing.  A  body  at  rest,  free  from 
external  influence,  would  remain  at  rest  forever.  In  other  words, 

(27) 


28  GEOMETKY   AND    FAITH. 

the  first  result  from  the  scientific  observation  of  motion  in  matter 
is,  that  matter  cannot  move.  Hence  follows  the  inevitable  con- 
clusion, that  the  cause  of  all  the  motion  in  the  universe,  is  some- 
thing else  than  matter.  Higher  than  this  the  investigation  of 
motion  itself  cannot  lead  us ;  but  this  is  high  enough  for  a  most 
valuable  stepping-stone. 

Why  do  we  ask  the  cause  of  motion  ?  Whence  do  we  derive 
the  idea  that  there  is  a  cause  for  it  ?  It  is  not  simply  the  impos- 
sibility of  our  imagining  a  beginning  ;  the  beginning  of  motion  we 
often  see.  But  the  motions  which  we  most  narrowly  examine  are 
those  produced  by  our  own  will ;  we  are  conscious  that  our  own 
volition  is  the  cause  of  such  motions ;  and  this  consciousness  is  the 
foundation  of  our  faith  that  motion  always  has  a  cause.  Is  this 
foundation  trustworthy  ?  Beyond  all  question  it  is.  Nay,  it  is 
the  foundation  of  all  possible  physical  science  ;  no  man  can  ex- 
tend a  generalization  beyond  the  particular  instances  for  which  he 
drew  it,  unless  he  leans  on  this  consciousness  of  causing.  To 
return  to  motion,  —  matter  cannot  move,  our  will  can  move  it ; 
there  is  nothing  to  suggest  any  other  origin  for  motion  than  voli- 
tion ;  hence  we  naturally,  and  legitimately,  infer  that  the  motion 
which  we  see,  everywhere  in  the  universe,  is  produced  by  a  will, 
independent  of  matter,  and  superior  to  all  the  phenomena. 

Thus  the  first  law  of  motion,  established  in  the  earliest  revival 
of  science,  demonstrates  not  only  the  existence  of  God,  but  his 
perpetual  presence  and  action.  Every  moving  thing  in  the 
heavens,  or  on  the  earth,  bears  the  same  sort  of  testimony  to  his 
being  and  presence,  as  that  borne  by  the  human  voiqe^jyid  action 
to  the  presence  of  a  man.  Whenever  we  see  anything  in  motion, 
God  is  the  mover.  In  the  ancient  tongues  this  was  one  of  his 
names.  The  winds  blow  at  his  command,  the  sun  rises  because  it 


MOTION.  29 

is  his  will,  the  falling  rain  and  running  stream  are  his  gift ;  and 
each  beating  pulse,  each  breath  that  we  unconsciously  draw,  is 
a  proof  that  this  machine  of  the  body  is,  each  moment,  depend- 
ent on  the  sustaining  love  and  power  of  its  Creator. 

Since  we  thus  refer  all  motion,  even  that  in  our  own  frames,  to 
the  will  of  God,  it  may  be  thought  that  we  are  destroying  man's 
freedom, —  making  him  a  mere  machine,  kept  in  motion  by  the 
Maker's  supervision.  Bht  this  objection  to  the  doctrine  of  man's 
present  dependence,  forgets  that  the  consciousness  of  our  freedom 
is  the  very  basis  on  which  AVC  have  built  our  faith  in  the  existence 
of  God.  It  is  from  our  own  consciousness  of  power,  to  cause  mo- 
tion at  our  own  will,  that  when  the  first  law  of  motion  has  ex- 
cluded us  from  ascribing  it  to  powers  inherent  in  matter  itself,  we 
ascribe  all  motion  to  his  will,  rather  than  to  any  unconscious 
natures. 

This  consciousness  of  our  own  power,  our  own  will,  may  be  de- 
nied in  words ;  but  it  will  presently  betray  itself,  lurking  in  the 
mind ;  it  cannot  be  really  denied  ;  it  is  the  foundation  of  all  phi- 
losophy and  faith.  The  body,  in  all  its  molecular  changes,  by 
which  ultimately  the  free  movement  of  the  limbs  is  produced,  is 
moved  by  the  will  and  power  of  God ;  the  first  law  of  motion 
proves  that ;  yet  the  direction  of  the  movement  in  the  limbs  is 
with  man,  consciousness  testifies  directly  to  this ;  man  is  free,  and 
cannot  heartily  believe  himself  to  be  otherwise. 

Our  muscular  power  is  not  ours,  but  it  is,  to  a  certain  extent, 
under  our  control.  We  cannot  lift  a  finger  without  the  aid  of  him 
who  formed  us ;  and  yet  it  is  we  who  move  our  hands.  So  the 
engineman,  who  has  not,  in  his  own  muscles,  strength  to  drive  a 
single  loom,  yet,  by  controlling  the  valves  of  his  engine,  keeps  the 
machinery  of  many  spindles  and  looms  in  motion.  Thus,  with  all 


30  GEOMETKY  AND    FAITH. 

man's  frailty,  and  his  absolute  dependence  upon  other  powers,  he 
yet  remains  a  cause,  —  free  and  efficient  to  control  and  direct  the 
engine  of  his  body,  wonderfully  framed  and  intrusted  to  his  care. 

Our  argument  has  been,  thus  far,  drawn  only  from  the  uniform 
velocity  of  motion ;  but  the  second  clause  in  the  first  law  would 
lead  to  the  same  result.  A  moving  body,  free  from  external  in- 
fluence, moves,  not  only  with  uniform  velocity,  but  in  a  straight 
line  forever. 

As  we  have  no  apparent  examples  in  nature  of  a  uniform  veloc- 
ity, so  we  have  none  of  uniform  direction.  External  influences 
perpetually  accelerate  or  retain  the  velocity  and  change  the  direc- 
tion of  moving  bodies.  But  as  the  first  part  of  the  law  is  derived, 
not  from  actual  examples  or  instances,  yet  by  irresistible  induction 
from  observed  facts,  so  the  second  part  follows  by  like  unavoida- 
ble inference  from  phenomena ;  indeed,  both  parts  are  defended, 
by  some  mechanicians,  as  axioms,  needing  no  other  proof  than 
Leibnitz's  principle  of  the  sufficient  reason. 

As  the  varied  motions  of  the  universe  cannot  have  sprung  from 
the  action  of  matter,  that  being  inert,  so  the  constant  changes 
of  direction  in  the  motions  prove  that  the  forces,  independent  of 
matter,  are  still  acting.  The  rebounding  of  a  solid  from  a  solid 
shows  that  the  particles  of  the  solid  adhere  by  some  form  of  force 
different  from  a  cohesion  of  contact,  —  elasticity  implies  that  the 
particles  are  held  together  by  some  force  which  permits  their  dis- 
tances from  each  other  to  vary  within  certain  limits.  When  the  ball 
leaves  the  muzzle  of  the  gun  its  path  instantly  begins  to  be  con- 
cave toward  the  earth  ;  and  would  be  so  at  any  conceivable  degree 
of  velocity.  The  meteor  passing  the  earth  at  eighty  miles  a 
second  bows  to  her  as  he  passes.  Thus  the  moon  also  is  perpet- 
ually deflected  from  its  path  by  the  earth,  and  the  earth  by  the 


MOTION.  31 

moon,  and  both  are  turned  constantly  aside  from  their  straight 
course  by  the  sun  ;  and  the  whole  host  of  heaven  is  constantly 
moving  in  a  rhythmic  dance,  wherein  each  star  influences  the 
motions  of  the  whole,  and  is  influenced  by  the  movements  of 
each  of  the  others. 

Our  consciousness  that  we  cause  motion  leads  us  to  ascribe  all 
change  of  velocity  to  force,  all  force  to  will.  The  same  conscious- 
ness bears  witness  also  that  all  change  of  direction  implies  the 
influence  of  will.  The  weight  of  bodies,  the  attraction  of  gravita- 
tion, the  correlated  forces  of  the  universe,  these  are  but  reverent 
forms  of  words  in  which  we  speak  of  that  which  can  only  be  re- 
ferred to  the  Divine  Will.  The  untaught  man,  the  poets  of  the 
earlier  ages,  were  more  true  to  reality  when  they  used  more  re- 
ligious forms  of  speech.  It  is  not  so  much  figurative,  as  literally 
true,  to  say  that  He  who  formed  the  Seven  Stars  and  Orion 
still  guides  them  on  their  way.  Their  circling  orbits  by  their 
figure,  and  the  golden  orbs  themselves  by  their  motion,  continu- 
ally manifest  the  presence  of  His  guiding  hand.  The  forces 
of  cohesion  and  repulsion,  of  electrical  and  chemical  change,  of 
heat,  of  light,  —  all  of  the  forces  by  which  the  existence  of  a 
particle  of  matter  can  possibly  make  itself  known  to  our  human 
senses,  are  but  manifestations  of  the  living  action  of  the  Most 
High. 

Thus  the  first  law  of  motion  leads  us  to  see  God  in  all  things, 
and  all  things  as  the  present  creations  of  his  hand.  It  might 
lead  us  astray,  it  might  lead  us  to  Pantheism,  were  it  not  that  it 
first  leads  us  to  perceive  that  force  is  an  attribute  of  will,  and  in- 
dependent of  matter ;  thus  keeping  us  to  the  conclusion  that  the 
Creator  and  Governor  of  all  things  is  free,  living,  —  and  our 
hearts  add,  good. 


VII. 

MUSCULAR  ACTION. 

WE  have  spoken  of  the  human  frame  as  an  engine  of  wonder- 
ful construction,  whose  movements  are  made  dependent  on  the 
human  will.  Yet  it  is  manifest  that  more  of  its  motions  are  in- 
dependent of  the  will  than  are  dependent  upon  it.  The  involun- 
tary muscles,  and  the  involuntary  movements  not  muscular,  are 
those  which  are  essential  to  the  very  existence  of  the  body.  The 
circulation  of  the  blood,  and  its  purification  through  the  alternat- 
ing expansion  and  contraction  of  the  chest,  are  obvious  instances 
of  these  vital,  involuntary  actions.  Not  less  important  is  each  one 
of  a  thousand  hidden  operations,  —  capillary  movements,  glandular 
secretions,  the  removal  of  the  effete  and  the  replacing  of  the  liv- 
ing molecules  ;  to  say  nothing  of  more  muscular  actions,  —  the 
peristaltic  motions,  and  the  wonderful  unconscious  artifices  of 
swallowing,  coughing,  sneezing,  and  the  like. 

The  voluntary  muscles  are  also  capable  of  involuntary  action. 
This  is  shown  not  only  by  occasional  convulsive  twitchings,  or 
more  violent  convulsions,  but,  in  a  still  more  instructive  manner, 
by  inveterate  habit,  operating  in  sleep,  or  even  when  the  will  op- 
poses. But,  although  the  action  which  has  become  habitual  is  not 
done  from  distinct,  conscious  volition,  the  habit  is  originally  formed 
by  acts  in  obedience  to  the  will.  The  law  by  which  the  voluntary 
action  becomes  involuntary  habit,  although  it  reduces,  too  fre- 
quently, man  to  slavery,  is  truly  beneficent  in  its  design,  and  in 

(32) 


MUSCULAR   ACTION.  33 

its  best  effects.  The  simplest  statement  of  the  law  would,  perhaps, 
be  found  in  saying  that  actions  which  have  been  associated  in  vo- 
lition become  associated  in  execution.  In  other  words,  when  we 
have  done  several  things  at  the  same  time,  or  in  quick  succession, 
the  attempt  to  repeat  one  of  these  actions  will  tend  to  produce  an 
involuntary  repetition  of  the  others.  For  an  illustration  of  the 
beneficent  action  of  this  law,  we  may  take  the  skillful  player  upon 
a  musical  instrument,  who  is  conscious  of  a  volition  only  at  the 
commencement  of  each  musical  phrase  :  the  fingering  of  the  sep- 
arate notes  comes  from  associated  execution.  In  a  familiar  piece 
his  volitions  would  be  even  less  frequent,  being  necessary  only  at 
the  commencement  of  a  new  strain. 

This  case  of  the  skillful  player,  being  less  frequent,  seems  the 
more  striking ;  but  there  is  scarcely  an  action  in  life  which  is  not 
aided  by  the  beneficent  operation  of  the  same  law,  just  as  there  is 
scarcely  a  mental  action  which  does  not  illustrate  the  kindred  law 
of  the  association  of  ideas.  The  child,  just  learning  to  walk, 
makes  a  painful  effort  at  each  successive  movement  of  each  mus- 
cle called  into  action.  It  requires  all  the  concentrated  energy  of 
his  will  to  make  the  successive  volitions  necessary  for  simple  step- 
ping from  chair  to  chair.  But  in  a  few  months  he  is  able,  by  as- 
sociated execution,  to  set  in  action,  by  a  single  volition,  a  series  of 
alternate  motions,  that  carry  him  forward,  without  his  attention,  in 
a  given  course,  at  a  uniform  speed.  No  power  of  will  is  required 
in  walking,  except  when  we  wish  to  alter  the  velocity,  or  the  direc- 
tion, of  our  movement. 

When  the  successive  movements,  dependent  on  associated  ex- 
ecution, are  connected,  as  in  walking,  by  a  law  of  simple  alterna- 
tion, the  case  is  not  difficult  of  explanation ;  and  the  physiologists 
show  us  how  the  will  relieves  itself  from  duty  by  a  switch,  turning 


34  GEOMETRY  AND    FAITH. 

off  the  currents  of  sensation  and  command  from  entering  the  main 
offi3e  in  the  brain. 

Other  cases,  in  which  the  operation  of  the  law  is  no  less  impor- 
tant to  our  comfort  and  convenience,  require,  however,  much  more 
intricate  combinations  of  movement.  In  many  familiar  occupa- 
tions we  require  our  hands  to  guide  an  instrument  rapidly  and 
freely  through  outlines  of  complex,  but  definite,  forms  ;  as,  for  ex- 
ample, in  writing,  or  in  free-hand  drawing.  All  men  have  more  or 
less  of  this  power  to  execute  ideal  figures,  or  to  imitate  given 
forms.  This  power  has  been  gained,  like  that  of  walking,  only 
through  repeated  and  laborious  efforts.  When  moving  the  hand 
in  one  direction,  we  need  a  new  volition  to  change  its  direction, 
or  to  alter  its  velocity ;  hence  our  first  attempts  at  curvilinear  mo- 
tion produce  polygonal  lines.  In  order  to  produce  a  curve,  as  in 
the  ordinary  forms  of  the  capital  letters,  we  must  produce,  by  sev- 
eral muscles  acting  at  once,  motions  in  several  directions  at  the 
same  time,  each  movement  varying  in  velocity  according  to  defi- 
nite laws.  To  draw,  for  example,  a  circle,  by  any  conceivable  set 
of  muscles  operating  on  the  arm,  would  require  at  least  three  sets 
of  muscles,  each  acting  in  a  different  direction,  and  no  two  ex- 
actly opposed ;  and  these  would  be  obliged  to  accelerate  and  re- 
tard their  action  by  peculiar  laws.  The  circle,  however,  is  the 
simplest  of  all  curves :  in  the  ordinary  operations  of  writing  and 
drawing,  the  rates  of  acceleration  and  retardation  must  follow 
more  complicated  laws.  Of  these  laws  we  think  nothing,  we  know 
nothing  :  we  see  the  curve  which  we  would  form,  and  a  single  im- 
pulse of  the  will  sends  the  pencil  along  the  waving  outline. 

Charles  Babbage,  a  successor  of  Sir  Isaac  Newton  in  the 
Lucasian  chair,  has  won  an  immortality  of  fame  by  inventing  a 
machin  3  which  will  tabulate  in  numbers  the  results  of  any  alge- 


MUSCULAR   ACTION.  36 

braic  law  which  it  may  be  set  to  obey.  But  how  much  mjre  won- 
derful is  this  calculating  engine  of  the  human  body,  which  is  not 
confined  to  arithmetical  results,  nor  does  it  require  that  its  director 
should  be  learned  in  algebraical  notation  to  set  it  at  its  appointed 
task,  but  which  is  set  by  the  artist,  with  his  delicate  perception  of 
the  beauty  of  form,  to  embody  his  divine  ideal !  and  it  obeys,  and 
places  before  us  on  the  canvas,  those  figures,  which,  unconsciously 
fulfilling  algebraic  or  numerical  law,  reach  far  higher,  and  express 
the  spiritual  thoughts  and  purposes  of  the  Master.  Is  not  the 
Maker  of  this  wondrous  engine  of  the  human  body  worthy  of  grat- 
itude and  adoration  V 

Consider  how  wonderful  is  the  phenomenon  of  a  boy's  throwing, 
successfully,  at  a  mark.  The  epicycloidal  theories  of  Hipparchus, 
the  Newtonian  theory  of  gravitation,  the  resolution  of  centripetal 
and  centrifugal  forces,  the  conic  sections  of  Apollonius,  the  modi- 
fications of  those  curves  by  the  resistance  of  the  air,  —  all  these 
are  involved  in  the  problem,  and  must  be  practically  solved,  with 
considerable  accuracy,  before  the  school-boy  can  give  his  fellow  a 
good  ball,  or  catch  one  on  the  fly. 

It  may  be  observed  that  the  mechanical  contrivance  by  which 
the  human  hand  is  enabled  to  go  through  all  imaginable  motions, 
and  strike,  at  a  free  sweep,  any  curve,  however  complicated  or 
however  beautiful,  is  an  embodiment  of  one  of  the  most  celebrated 
of  mathematical  conceptions,  discussed  in  the  writings  of  Plato 
and  Aristotle,  constituting,  in  its  development,  one  of  the  chief 
triumphs  of  Hipparcbus,  and  brought  by  modern  mathematicians, 
through  the  arithmetic  of  sines,  and  the  canon  mirificus  of  Napier, 
into  a  form  capable  of  reducing  to  a  regular  curve  the  most  vari- 
able and  irregular  table  of  observations.  In  this  method  of  epi- 
cycles, as  used  by  the  modern  computer,  a  series  of  arms  is  sup- 


86  GEOMETRY  AND   FAITH. 

posed  to  be  carried,  each  on  the  extremity  of  the  preceding,  and, 
during  the  revolution  of  the  first,  each  to  revolve  once  oftener  than 
the  preceding ;  that  is,  while  the  first  arm  of  the  series  revolves 
once,  the  second  revolves  twice,  the  third  three  times,  and  so  on. 
It  only  remains  for  the  computer  to  fix  the  length  of  these  arms, 
and  determine  their  original  position,  in  order  to  make  the  end  of 
the  third  or  fourth,  or,  in  cases  of  difficulty,  of  the  fifth  and  sixth, 
describe  any  path  he  wishes.  In  the  human  limb,  the  upper  arm 
is  the  first,  the  fore-arm  the  second,  the  hand  the  third,  and  the 
fingers  the  fourth,  fifth,  and  sixth  of  these  rotating  arms ;  and 
the  fixedness  in  the  ratio  of  their  length  is  more  than  compensated 
for,  by  our  ability  to  graduate  the  ratio  of  their  revolution  at  will. 
Is  there  no  meaning  in  the  fact  that  the  most  cunning  device  of 
human  ingenuity  for  making  a  point  travel,  under  simple  laws, 
through  the  greatest  variety  of  paths,  should  thus  prove  to  be  sub- 
stantially the  same  with  that  adopted  in  the  very  creation  of  the 
human  frame,  for  enabling  the  hand  to  guide  its  tools  with  freedom 
and  accuracy  ? 


VIII. 

GEOMETRICAL   INSTINCTS. 

SINCE  our  fellow  citizens  of  all  the  animal  kingdom  are,  like 
ourselves,  dwellers  in  space  and  time,  it  is  necessary  for  them, 
also,  to  have  ideas  of  distance  and  direction  in  space,  duration 
and  lapse  in  time.  Ideas  gained  by  sense-perception  seem  to  fur- 
nish them,  as  us,  the  data  for  reasoning  ;  but  ideas  of  direct  intui- 
tion do  not  appear  to  afford  to  them,  as  to  us,  objects  whereon  to 
reason  ;  but  merely  serve,  as  certain  of  the  kind  do  for  us,  as  the 
stimulus  of  desire,  and  the  guide  of  unreflective  action.  These 
intuitive  ideas,  perceived  by  inward  sense,  but  not,  perhaps,  dis- 
tinctly eliminated  in  consciousness  from  co-existent  ideas,  are,  in 
the  lower  animals,  called  instincts  ;  and  when  used  in  like  manner 
by  us,  not  as  propositions  for  conscious  reasoning,  but  as  the 
grounds  of  instantaneous  judgment  or  action,  they  have  the  vari- 
ous names  of  instincts,  feelings,  promptings,  conscience,  or  genius, 
according  to  the  nature  of  the  objects  to  which  they  relate. 

Geometrical  instincts  are  common  to  us  with  all  the  animate 
races.  That  instinctive  trigonometry,  for  instance,  by  which  a 
child,  of  a  few  months  old,  learns  to  tell  the  position  of  any  object, 
to  which  his  two  eyes  are  directed,  is  probably  exercised  by  all 
animals  with  two  eyes  capable  of  being  turned  upon  a  single  object. 
The  most  striking  instance  is  popularly  believed  to  be  the  young 
quail,  which  is  said  to  run,  as  soon  as  hatched,  freely  about,  peck- 
ing at  minute  objects  with  as  true  an  aim  as  its  mother's.  I  have 

(37) 


38  GEOMETKY  AND   FAITH. 

seen  a  Setter's  pup,  sired  by  a  Pointer,  when  a  few  weeks  old, 
point  at  a  piece  of  anthracite  with  all  the  accuracy  of  its  father, 
which  it  had  never  seen.  In  1843,  a  toad,  frequenting  the  gar- 
den at  Burgoyne's  Headquarters,  in  Cambridge,  and  losing  by  an 
accident  the  sight  of  one  eye,  was  for  a  long  time  unable  to  aim 
his  tongue,  with  certainty,  at  the  overloaded  bees,  who,  returning, 
missed  the  threshold  of  the  hive,  under  which  the  toad,  expecting 
such  misfortunes  of  his  insect  neighbors,  was  accustomed  to  sit  and 
await  their  fall.  In  time  he,  like  human  beings  who  have  had  like 
accidents  befall  them,  learned  to  substitute  optics  for  trigonometry, 
and  instead  of  solving  triangles,  with  a  base  and  adjacent  angles 
given,  decided  on  the  position  of  objects  requiring  a  certain  focal 
adjustment  and  direction  of  his  remaining  telescope. 

But  how  complicated  the  action  by  which  he  proved  this,  if  I 
may  use  the  expression,  unconscious  knowledge  of  the  position  of 
the  bee  !  A  single  conscious  volition,  and  his  tongue,  which  is 
rooted  in  the  front  of  his  mouth,  with  the  tip  lying  far  down  within 
his  throat,  flies  out  and  back  like  an  electric  spark,  having  taken 
the  bee  up  on  its  tip,  and  thrust  her  down  the  throat  of  the  toad. 
His  calculating  engine,  set  by  the  adjustment  of  his  eyes,  not  only 
computes  the  exact  curve  in  which  the  tip  of  his  tongue  must 
move,  but  the  exact  force  and  velocity  with  which  it  must  be 
sent,  in  order  to  accomplish  its  mechanical  errand.  The  same 
marvelous  unconsriious  calculation  is  proved  when  a  boy  hits  the 
mark,  at  which  he  aims  a  stone,  or  when  the  expert  player  at  bill- 
iards strikes  his  ball  on  exactly  the  right  part  of  the  ball,  in 
exactly  the  right  direction,  and  with  exactly  the  right  force,  in 
order  to  make  it  pursue  a  long  course,  partly  curved  and  partly 
straight,  with  rebounds  from  the  cushion,  and  rebounds  from  other 
balls,  and  come  to  rest  at  a  determined  place.  He  does  not  know 


GEOMETRICAL   INSTINCTS.  39 

the  difficulties  of  the  problem  he  has  solved  ;  he  does  it  with  as 
little  of  conscious  calculation  as  that  with  which  the  toad  snaps 
up  the  bee ;  but  this  only  renders  the  more  striking  the  wonderful 
perfection  of  the  muscular  and  nervous  organization,  as  machinery 
adapted  to  describe  geometrical  figures  and  solve  mechanical  prob- 
lems of  great  perplexity. 

The  architectural  or  nest-building  instincts  of  animals  show  the 
geometrical  and  mechanical  knowledge  of  the  Creator  of  animals 
in  a  very  conspicuous  manner.  Men  invented  and  used  the  arch 
long  before  human  mathematicians  solved  its  theory.  Many  other 
of  our  mechanical  inventions,  and  some  of  them,  as  the  barrel  and 
the  potter's  wheel,  for  example,  of  a  wonderful  kind,  have  an 
antiquity  that  long  antedates  abstract  mathematical  thought.  We, 
reasoning,  discover  the  principles  underlying  our  inventions,  and 
thus  improve  science,  which  again  suggests  new  inventions ;  so 
that  human  art  and  human  science  stimulate  and  foster  each  other 
to  endless  competition  and  endless  progress.  The  lower  races  have, 
apparently,  no  abstract  thoughts,  no  intuitions,  that  are  brought, 
with  consciousness,  among  the  data  of  their  reasoning ;  in  other 
words,  they  appear  to  have  no  science,  and  hence  their  progress 
in  the  arts  is  so  slow  as  to  appear  stationary.  But  their  instinc- 
tive judgments  appear,  frequently,  more  accurate  and  wonderful 
than  those  of  men.  To  see  the  republican  swallow,  coming  through 
the  air,  fold  her  wings  at  precisely  the  right  moment,  and  when  at 
precisely  the  right  speed,  in  order  to  enter  softly  and  smoothly 
her  earthen  bottle,  makes  the  art  of  the  most  skillful  coxswain 
seem  rude  in  comparison.  The  weaving  of  the  bird's  nest  is  in  the 
case  of  the  African  grosbeak  carried  to  a  degree  of  perfection  that 
vies  with  that  of  the  nicest  works  of  man,  unaided  by  machinery. 

But  the  architectural  work  of  insects  is  most  wonderful,  and 


40  GEOMETRY   AND    FAITH. 

none  more  so  than  the  familiar  honeycomb.  Always  admired  by 
men,  from  the  earliest  ages,  it  was,  at  the  beginning  of  the  last 
century,  discovered  by  Maraldi,  of  Nice,  to  embody  distinctly  the 
complicated  geometrical  conception,  of  forming  cells  to  contain  a 
fluid  mass,  with  the  greatest  strength,  the  greatest  economy  of 
space,  and  the  greatest  economy  of  material.  The  paper-making 
wasps  make  rude  approximations  toward  the  solution  of  the  same 
problem,  but  inasmuch  as  the  larger  part  of  their  material  is 
cheaply  gathered  from  the  surface  of  wood,  there  is  no  call  for  so 
strict  an  economy.  The  bee,  needing  a  water-proof  material,  yet 
finding  the  resin  of  trees  too  adhesive  to  be  worked  with  facility, 
confines  her  use, of  such  resin  to  the  places  in  which  she  needs 
especial  strength,  or  especial  resistance  to  moisture  ;  and,  for  her 
ordinary  work  of  cell  building,  uses  a  material  wholly  secreted  by 
the  glands  of  her  own  body.  Her  cells  are  approximately  hexa- 
gons, which  hold  more,  and  have  shorter,  and  therefore  stronger, 
sides  than  any  other  figures  which  could  be  packed  without  waste 
of  room.  They  are  set,  with  economy  of  room,  base  to  base  ;  and 
still  further  strengthened  on  the  bases,  by  being  set  one  against  a 
part  of  three,  so  that  the  bottom  of  each  cell  is  suuported  by  three 
partition  walls  on  the  other  side.  Finally,  and  most  curious,  the 
bottom  of  each  cell  is  depressed  in  the  centre  to  about  that  degree 
which  will  save  most  by  diminishing  the  height  of  these  supporting 
partitions  without  increasing  too  much  the  area  of  the  floor  which 
rests  upon  them.  I  say  about  that  degree  ;  and  the  accordance 
of  the  average  cells,  in  a  normal  piece  of  comb,  with  theoretical 
perfection  as  determined  by  the  calculus  of  Newton,  is  very  close. 
We  should  not  expect  perfection,  because  the  perfection  of  the 
artisan  is  to  be  measured,  not  by  the  perfection  of  his  results,  but 
by  the  perfection  of  their  adaptation  to  his  end.  He  were  a  poor 


GEOMETRICAL   INSTINCTS.  41 

farrier  who  polished  his  shoes  with  the  care  that  a  dentist  bestows 
upon  his  gold  filling.  Nor  would  the  bee  be  a  wise  economist  if 
she  wasted  time  in  bringing  to  theoretical  perfection  her  saving  of 
wax.  What  the  bee's  conscious  aim  is,  in  the  construction  of  the 
cell,  we  may  or  may  not  at  some  time  discover.  That  she  has  con- 
scious aims  is  evident,  from  her  adaptation  of  the  form  of  the 
comb  to  circumstances,  and  from  her  ingenious  contrivances,  not 
only  to  repair  mischief,  but  to  guard  against  threatened  evil.  But 
it  is  equally  evident  that  in  the  formation  of  the  bee,  and  in  the 
inspiration  of  her  instincts,  a  knowledge  and  wisdom  presided,  to 
which  the  whole  question  of  maximum  and  minimum  lay  open 
countless  ages  before  human  thought  grappled  with  its  problems. 

It  only  requires  a  more  intimate  acquaintance  with  the  habits 
of  any  animal  to  discover  the  adaptation  of  its  instincts  to  its  or- 
ganization. The  apparent  instances  to  the  contrary  arise  from 
want  of  patience  and  thoughtfulness  in  the  observer.  I  stood,  one 
evening,  at  early  dusk,  watching  the  movements  of  a  curious  insect 
on  the  inside  wall  of  an  open  shed.  Its  body,  a  little  over  an  inch 
in  length,  and  very  thin,  seemed,  nevertheless,  too  heavy  for  its 
long  and  delicate  legs,  which  swayed  and  trembled  under  the 
weight,  as  it  slowly  stepped  along,  with  long  pauses  between  each 
step.  It  walked  on  four  legs,  holding  the  other  two,  which  were 
shorter  and  stouter,  extended  in  front.  I  presently  perceived  that 
it  was  making  toward  a  fly  which  had  settled,  apparently  to  sleep, 
upon  the  board  within  three  inches  of  my  insect.  I  wished  to  see 
what  its  designs  were  upon  the  fly,  but  so  slow  were  its  motions, 
that  I  was  obliged  to  wait  fully  twenty  minutes  before  being  grati- 
fied. As  the  insect  approached  the  fly,  he  slowly  extended  a  very 
long  and  exceedingly  slender  antenna,  and  touched  the  fly  gently, 
in  various  parts,  as  if  to  ascertain  more  precisely  its  position.  He 


42  GEOMETRY  AND   FAITH. 

then  made  a  detour,  and  brought  himself,  at  length,  exactly  in 
front  of  the  sleeping  victim,  with  his  own  head  nearly  over  the  fly's 
head,  and  began  very  slowly  to  raise  his  raptorious  legs  high  above 
the  fly.  I  was  growing  tired  of  his  slow  and  awkward  motions, 
when,  in  an  instant,  my  feelings  were  changed  to  those  of  the 
highest  admiration  for  his  great  engineering  skill ;  the  fly  was 
aloft  in  air,  with  the  beak  of  the  insect  thrust  into  its  back  calmly 
imbibing  its  juices  ;  while  the  fly's  feet  could  touch  nothing,  its 
wings  were  both  dislocated,  and  firmly  pinioned  in  the  captor's 
raptorious  legs ;  which,  coming  down  suddenly  between  the  wings, 
had  parted  them,  dislocated  them,  and  pinned  them  between  the 
wrist-spurs  of  those  legs  and  their  sharp  extremities ;  then,  with- 
out an  instant's  pause,  lifted  the  fly  from  his  feet,  and  impaled 
him  upon  the  ploiaria's  beak. 


IX. 

MOTION  ETERNAL  IN  DURATION. 

THE  motion  of  bodies  is  not  observed  to  be  with  uniform  ve- 
locity. We  see  bodies  at  rest  beginning  to  move,  and  bodies  in 
motion  coming  to  rest.  Let  us  consider  these  cases  a  little  more 
closely. 

We  have  said  that  motion  implies  force,  and  that  force  implies 
will.  Force  is  the  energy  of  will  acting  upon  matter.  But  how 
does  the  will  affect  matter  which  is  foreign  to  will,  and  over  which 
will,  we  might  therefore  suppose,  would  have  no  control  ?  To  this 
question  we  answer  that  the  human  will  never  affects  the  material 
thing  which  it  determines  to  move,  except  through  the  agency  of 
material  agents  of  whose  existence  and  motions  it  may  be  uncon- 
scious. Not  to  speak  of  the  unconsciousness  of  all  earnest  labor, 
the  absorption  of  the  mind  in  its  object,  take  the  case  in  which 
the  mind  is  seeking  to  analyze  its  volitions,  and  it  will  be  found 
that  we  cannot  reach,  in  our  analysis,  the  first  effect  of  the  volition 
upon  the  physical  frame.  In  the  movements  of  my  hand,  although 
I  know  it  is  effected  by  the  movement  of  certain  muscles,  and 
that  this  is  effected  through  the  nerves  of  volition,  yet  I  cannot 
trace  my  will  behind  the  command  issued  to  the  hand  itself. 
Hence  it  may  be  said  that  the  human  will  moves  even  the  human 
body  through  physical  agencies,  and  the  manner  of  its  control  over 
these  agencies  is  known  only  by  Him  who  breathed  into  us  the 
breath  of  life. 

'43) 


44  GEOMETRY  AND   FAITH. 

In  like  manner  is  it  with  the  natural  motions  which  we  see  on 
every  side.  All  bodies  are  moved  through  the  agency  of  other 
bodies,  and  we  see  nowhere  a  motion  which  is  not  dependent  upon 
physical  causes,  that  is,  which  is  not  produced  by  physical  agents. 
Doubtless,  He  by  whose  will  all  things  are  moved  is  not  restricted 
from  any  mode  of  action,  and  He  can  move  bodies  independently 
of  all  law,  and  without  any  intervention  of  means.  Nevertheless, 
such  motion  would  be  miraculous,  and  out  of  the  course  of  nature. 
Our  will  employs,  unconsciously,  the  aid  of  nerve  and  muscle  ;  the 
Supreme  Will  employs,  with  wise  designs,  the  intervention  of  the 
laws  of  impulse,  attraction,  and  repulsion. 

But  when  a  body  at  rest  receives  motion  through  impulse,  it 
evidently  continues  the  motion  of  the  impelling  body.  So  that,  if 
the  impelling  body  is  put  to  rest  by  its  contact,  the  motion  is  not 
lost,  but  only  transferred.  And  this  is  true  independently  of  the 
size  of  the  two  bodies.  The  earth  is  not  immovable,  and  the  small- 
est grain  of  dust  that  falls  upon  it  strikes  with  a  certain  amount  of 
force. 

Again  :  when  bodies  act  upon  each  other  by  attraction  or  repul- 
sion, the  force  acts  upon  both  bodies.  As  surely  as  two  vessels, 
floating  on  still  water,  would  both  move  when  one  attempted  to 
draw  or  push  the  other,  so  surely  must  each  moving  thing  move 
those  attracted  by  it,  and  all  that  attract  it. 

This  motion,  also,  exists,  whatever  be  the  relative  size  of  the 
bodies.  In  the  case  of  motion  produced  by  direct  attraction,  — 
for  instance,  in  the  fall  of  bodies  toward  the  earth,  —  the  motion 
is  in  simple  inverse  proportion  to  the  masses.  It  is,  therefore, 
capable  of  easy  arithmetical  calculation.  The  weight  of  the  earth 
in  milligrams  may  be  nearly  represented  by  the  figure  6  followed 
by  thirty  cyphers,  or  six  nonillions.  Hence  the  falling  of  a  little 


MOTION   ETERNAL  IN   DURATION.  45 

insect,  weighing  six  milligrams,  would  move  the  earth  the  non- 
illionth  part  as  much  as  the  insect  fell.  That  is,  the  thirtieth 
place  of  decimals  is  capable  of  representing  the  motion  given  to 
the  earth  by  the  fall  of  the  smallest  bodies. 

There  is,  then,  in  nature,  no  provision  for  the  destruction  of 
mution,  but  only  for  its  transference.  Motion  in  a  body  free  from 
external  influences  is  uniform  in  velocity  and  direction ;  it  can  be 
retarded,  and  apparently  destroyed,  only  by  external  influences, 
by  impulse,  or  by  attraction  or  repulsion.  But  these,  we  have 
shown,  cannot  really  destroy ;  they  simply  transfer  the  motion  to 
the  interfering  body.  Hence  all  motion  is  eternal ;  it  is  communi- 
cated to  an  ever-increasing  amount  of  matter ;  it  is,  in  each  suc- 
cessive particle  affected,  less  in  quantity,  but  never  becomes  noth- 
ing, since  the  sum  of  the  motion  in  all  the  particles  remains  the 
same. 

It  is  sometimes  affirmed,  since  the  demonstration  of  the  theory 
that  heat  and  light  are  undulatory  motions,  that  all  mechanical 
motions  finally  take  the  form  of  heat,  or  of  light.  The  sound  of  a 
tuning-fork,  placed  upon  a  mass  of  caoutchouc,  is  inaudible,  and 
a  delicate  thermo-galvanic  test  shows  that  the  temperature  of  the 
rubber  has  been  raised.  But  a  still  more  delicate  test,  if  it  were 
possible  to  apply  one,  might  show  a  quality  in  this  heat  that  would 
demonstrate  it  to  have  arisen  from  a  fork ;  just  as  Clairault's  nicer 
calculation  showed  the  comet  of  1770  to  have  been,  before  it  had 
its  short  period  within  the  orbit  of  Jupiter,  a  wanderer  outside  the 
Jovian  sphere. 


X. 

MOTION  OMNIPRESENT  IN  SPACE. 

CHARLES  B  ABB  AGE,  in  the  "  Ninth  Bridge-water  Treatise,"  has 
a  chapter  concerning  the  permanent  impression  of  our  words  upon 
the  air,  —  a  chapter  which  none  have  ever  read  without  a  thrill  of 
mingled  admiration  and  fear ;  and  which  closes  with  an  eloquence 
that  were  worthy  the  lips  of  an  orator,  though  coming  from  a 
mathematician's  pen. 

Would  that  Babbage  had  touched,  in  his  fragmentary  treatise, 
upon  some  of  the  inferences  which  may  be  drawn  from  the  New- 
tonian law  of  gravity,  —  inferences  which  would  probably  have 
been  as  new  to  most  of  his  readers  as  those  which  he,  with  so 
much  acuteness,  draws  from  the  law  of  the  equality  of  action  and 
reaction. 

The  motion  of  which  Babbage  speaks,  in  the  chapter  to  which  we 
refer,  is  undulatory,  communicated  by  impulse,  and  requiring  time 
for  its  transmission  ;  and  the  startling  result  of  his  reasoning  comes 
from  the  never-dying  character  of  the  motion,  keeping  forever  a 
record  of  our  words  in  the  atmosphere  itself,  always  audible  to  a 
finer  sense  than  ours ;  reserved  against  the  day  of  account,  when, 
perchance,  our  own  ears  may  be  quickened  to  hear  our  own  words 
yet  ringing  in  the  air. 

But  motion  is  not  only  enduring  through  all  time,  it  is  simulta- 
neous throughout  all  space.  The  apple  which  falls  from  the  tree  is 
met  by  the  earth  :  not  half  way,  but  at  a  distance  fitly  proportioned 

(46) 


MOTION   OMNIPRESENT   IN   SPACE.  47 

to  their  respective  masses.  The  moon  follows  the  movement  of  the 
earth  with  instant  obedience,  and  the  sun  with  prompt  humility 
bends  his  course  to  theirs.  The  sister  planets  with  their  moons 
are  moved  by  sympathy  with  earth,  and  the  stars  and  most  distant 
clusters  of  the  universe  obey  the  leading  of  the  sun.  Thus  through- 
out all  the  fields  of  space,  wherever  stars  or  suns  are  scattered, 
they  move  for  the  falling  apple's  sake.  Nor  is  the  motion  slowly 
taken  up.  The  moon  waits  for  no  tardy  moving  impulse  from  the 
earth,  but  instantly  obeys.  The  speed  of  light  which  reaches  the 
sun  in  a  few  minutes  would  be  too  slow  to  compare  with  this. 
Electricity  itself,  coursing  round  the  earth  a  thousand  times  an 
hour,  can  give  us  no  conception  of  the  perfectly  simultaneous 
motions  of  gravity.  There  are  stars  visible  to  the  telescopic  eye, 
whose  light  has  been  ages  on  its  swift-winged  course  before  it 
reached  this  distant  part  of  space ;  but  they  move  in  instant 
accordance  with  the  falling  fruit. 

True  it  is,  that  our  senses  refuse  to  bear  witness  to  any  motion 
other  than  the  apple's  fall,  and  our  fingers  tire  if  we  attempt  to 
write  the  long  list  of  figures,  which  our  Arabic  notation  requires 
to  express  the  movement  thereby  given  to  the  sun.  Yet  that  mo- 
tion can  be  proved  to  exist,  and  the  algebraist's  formula  can  rep- 
resent its  quantity.  The  position  of  every  particle  of  matter  at 
every  instant  of  time,  past,  present,  or  to  come,  has  been  written 
in  one  short  sentence,  which  any  man  can  read.  And  as  each 
man  can  understand  more  or  less  of  this  formula  of  motion,  accord- 
ing to  his  ability  and  his  acquaintance  with  mathematical  learning, 
so  we  may  conceive  of  intelligent  beings,  whose  faculties  are  very 
far  short  of  infinite  perfection,  who  can  read  in  that  sentence  the 
motions  not  only  of  the  sun,  but  of  all  bodies  which  our  senses 
reveal  to  us.  Nay,  if  the  mind  of  Newton  has  advanced  in  power 


48  GEOMETRY  AND    FAITH. 

since  he  entered  heaven  with  a  speed  at  all  proportioned  to  his  in- 
tellectual growth  on  earth,  perhaps  even  he  could  now  with  great 
ease  assign  to  every  star  in  the  wide  universe  of  God  the  motion 
which  it  received  from  the  fall  of  that  apple  which  led  him  to  his 
immortal  discoveries. 

Every  moving  thing  on  the  earth,  from  the  least  unto  the  great- 
est, is  accompanied  in  motion  by  all  the  heavenly  spheres.  The 
rolling  planets  influence  each  other  on  their  path,  and  each  is  influ- 
enced by  the  changes  on  its  surface.  The  starry  systems,  wheel- 
ing round  their  unknown  centre,  move  in  harmony  with  each  other, 
and  bend  each  other's  courses,  and  each  is  moved  by  the  planets 
which  accompany  it  in  its  mighty  dance.  Thus  does  this  law  of 
gravitation  bind  all  material  bodies  in  one  well-balanced  system, 
wherein  not  one  particle  can  move  but  all  the  uncounted  series  of 
worlds  and  suns  must  simultaneously  move  with  it. 

Thus  may  every  deed  on  earth  be  instantly  known  in  the  farthest 
star,  whose  light,  traveling  with  almost  unbounded  speed  since  cre- 
ation's dawn,  has  not  yet  reached  our  eyes.  It  only  needs  in  that 
star  a  sense  quick  enough  to  perceive  the  motion,  infinitely  too 
small  for  human  sense,  and  an  analysis  far  reaching  enough  to 
trace  that  motion  to  its  cause.  The  cloud  of  witnesses  that  ever 
encompass  this  arena  of  our  mortal  life  may  need  no  near  approach 
to  earthly  scenes,  that  they  may  scan  our  conduct.  As  they  jour- 
ney from  star  to  star,  and  roam  through  the  unlimited  glories  of 
creation,  they  may  read,  in  the  motions  of  the  heavens  about  them, 
the  ever-faithful  report  of  the  deeds  of  men. 

This  sympathetic  movement  of  the  planets,  like  the  mechanical 
impulse  given  by  our  words  to  the  air,  is  everduring. 

The  astronomer,  from  the  present  motion  of  the  comet,  learns 
all  its  former  path,  traces  it  back  on  its  long  round  of  many  years, 


MOTION    OMNIPRESENT   IN    SPACE.  49 

shows  you  when  and  where  it  was  disturbed  in  its  course  by  plan- 
ets, and  points  to  you  the  altered  movement  which  it  assumed  from 
the  interference  of  bodies  unknown  by  any  other  means  to  human 
science.  He  needs  only  a  more  subtle  analysis,  and  a  wider  grasp 
of  mind,  to  do  for  the  planets  and  the  stars  what  he  has  done  for 
the  comet.  Nay,  it  were  a  task  easily  done  by  a  spirit  less  than 
infinite  to  read  in  the  present  motion  of  any  one  star  the  past  mo- 
ti  ••us  of  every  star  in  the  universe,  and  thus  of  every  planet  that 
wheels  round  those  stars,  and  of  every  moving  thing  upon  those 
planets. 

Thus  considered,  how  strange  a  record  does  the  star-gemmed 
vesture  of  the  night  present !  There,  in  the  seemingly  fixed  order 
of  those  blazing  sapphires,  is  a  living  dance,  in  whose  mazy  track 
is  written  the  record  of  all  the  motions  that  ever  men  or  nature 
made.  Had  we  the  skill  to  read  it,  we  should  there  find  written 
every  deed  of  kindness,  every  deed  of  guilt,  together  with  the  fall 
of  the  landslide,  the  play  of  the  fountain,  the  sporting  of  the  lamb, 
and  the  waving  of  the  grass.  Nay,  when  we  behold  the  super- 
human powers  of  calculation,  exhibited  sometimes  by  sickly  chil- 
dren, long  before  they  reach  man's  age,  may  we  not  believe  that 
men,  when  hereafter  freed  from  the  load  of  this  mortal  clay,  may 
be  able  in  the  movement  of  the  planets  or  the  sun  to  read  the  rec- 
ords of  their  own  past  life  ? 

Thou,  who  hast  raised  thy  hand  to  do  a  deed  of  wickedness,  stay 
thine  arm !  The  universe  will  be  witness  of  thine  act,  and  bear 
an  everlasting  testimony  against  thee ;  for  every  star  in  the  re- 
motest heavens  will  move  when  thy  hand  moves,  and  all  the  tear- 
ful prayers  thy  soul  can  utter  will  never  restore  those  moving  orbs 
to  the  path  from  which  thy  deed  has  drawn  them. 


XI. 

THE   SPHERE   OF   HUMAN  INFLUENCE. 

THE  conclusions  of  the  last  chapter  would  need  but  slight  modi- 
fication, should  any  future  observations  reveal  the  fact  that  the 
motions  of  gravity  are  not  absolutely  simultaneous.  If  gravity 
should  prove  to  be  a  mere  resultant  of  the  undulations  of  light 
and  heat,  we  should  gain,  indeed,  a  magnificent  illustration  of  the 
inspired  wisdom  which  begins  its  account  of  creation  with  recording 
the  fiat,  "  Let  there  be  light ;  "  but  we  should  not  lose  the  spir- 
itual lessons  drawn  from  the  fact  that  the  material  universe  is 
bound  by  gravitation  into  one  sensitively-balanced  whole,  so  that 
each  deed  of  man  is  felt  in  the  farthest  star,  and  a  perpetual  rec- 
ord thereof  is  kept  in  the  movement  of  the  heavenly  orbs. 

u  Every  natural  fact  is  a  symbol  of  some  spiritual  fact."  As 
motion  is  propagated  throughout  all  space,  and  endures  through  all 
time,  so  each  change  in  the  spirit  of  each  man  affects  the  state  of 
the  spiritual  universe,  and  its  influence  remains  through  all  eter- 
nity. As  matter  by  the  law  of  gravity,  so  spirits  by  a  law  of  sym- 
pathetic attraction  are  all  bound  in  one  harmonious  whole,  whereof 
"  if  one  member  suffer,  all  the  members  suffer  with  it."  Liebig's 
law,  that  a  moving  particle  communicates  its  motion  to  adjacent 
particles,  was  announced  in  defence  of  a  mistaken  theory  of  trichi- 
nosis, but  the  law  itself  is  true,  and  universal  in  physics,  in  physi- 
ology, and  in  psychology. 

The  law  of  attraction  holds  the  same  place  of  primary  impor- 

(50) 


THE   SPHERE   OF  HUMAN   INFLUENCE.  51 

tance  in  considering  man,  as  in  considering  matter.  Our  great 
economist  makes  the  Unity  of  Law  a  fact  of  primal  importance  in 
the  development  of  his  grand  and  cheering  theses.  The  eagle 
saint  of  the  Christian  Church  declares  that  God  is  love ;  and  all 
the  highest  religions  teach  that  man  is  made  in  God's  image.  He 
is  the  sun  of  infinite  magnitude,  the  origin  of  all  forces,  but  not 
moved  by  any  reaction  ;  we  are  the  particles  moved  by  Him  both 
immediately  and  mediately  through  each  other.  Love  is  the  funda- 
mental law  ;  the  sympathy  between  human  souls  is  always  greater 
than  the  antipathy  ;  even  when,  through  disturbing  forces,  the 
sympathy  is  for  a  time  neutralized,  and  the  antipathy  is  developed 
into  hatred. 

The  influence  which  a  man  exerts  does  not  cease  with  the  effect 
that  he  has  upon  his  most  intimate  friends  ;  nor  does  it  flow  from 
the  power  of  his  word  alone,  nor  from  the  mere  force  of  his 
example.  Whatever  a  man  does,  or  thinks  or  feels,  even  in  soli- 
tude, has  an  effect  upon  the  world.  For,  in  the  first  place,  it 
affects  himself  and  his  own  character;  and  that  character  must 
influence,  in  some  manner,  those  with  whom  he  comes  in  contact ; 
influence  them  in  proportion  to  the  strength  of  his  power  to  affect 
them,  and  to  the  weakness  of  their  power  to  resist  him.  A  cheer- 
ful countenance  carries  a  gleam  of  sunshine  into  the  darkest  alley  ; 
a  sad  face  throws  a  shadow  over  the  hearts  of  those  who  pass  it, 
even  on  a  crowded  thoroughfare  ;  thus  every  shade  of  thought  and 
feeling,  expressed  in  the  countenance,  or  in  word,  or  gesture,  or 
action,  produces  some  corresponding  change,  slight  though  it  may 
be,  in.  all  souls  that  recognize,  however  dimly,  the  expression. 
And  this  change  transfers  itself,  in  varying  proportions,  to  ever- 
widening  circles.  Thus  the  spirit  and  tone  of  the  age  is  the  sum 
of  the  individual  thoughts,  and  thus  also  the  individual  character 


52  GEOMETRY   AND    FAITH. 

of  each  man  is  to  some  extent  the  product  of  all  the  preceding 
ages  of  the  race. 

By  the  manners  of  a  man,  or  by  his  speech,  we  know  whether 
his  companions  have  been  Galileans  or  Athenians.  A  close  ob- 
server in  the  city  can  tell,  of  the  majority  of  strangers  he  may 
chance  to  notice,  their  age,  their  character,  their  calling,  the 
place  of  their  residence,  and  of  their  nativity,  or  that  of  their 
ancestors.  It  would  only  need  a  nicer  observation,  a  closer  in- 
sight, a  more  searching  analysis,  to  detect  in  a  stranger's  heart 
both  the  original  traits  of  his  character  and  the  modifications  due 
to  the  influence  of  all  with  whom  he  has  been  associated.  It  might 
be  a  task  no  more  above  a  Shakespeare's  grasp,  than  the  creation 
of  a  Hamlet  is  above  the  power  of  an  ordinary  man,  to  trace,  in  a 
man's  present  character,  the  influence  of  every  person  and  every 
circumstance  that  have  ever  acted  upon  him  to  repress  or  to  de- 
velop his  powers.  It  would  not  require  an  absolutely  infinite  intel- 
lect to  trace  the  effect  of  any  humble  act  of  an  honest  man,  until 
it  had  been  seen  to  have  blessed  millions  of  his  fellow-men  ;  nor  to 
show  the  loss  or  suffering  that  have  flowed  to  thousands  from  an 
evil  deed.  It  may  hereafter  be  possible  for  some  higher  intelli- 
gence than  ours  to  read  the  record  of  my  interior  life,  written  in 
positive  or  negative  characters,  upon  the  soul  of  some  poor  man 
whom  I  have  never  seen,  but  whom  I  must,  nevertheless,  have 
helped  or  hindered  by  my  every  act  and  word. 

Thus  the  spiritual  universe  is  bound,  by  the  law  of  love,  under 
its  wider  enunciation  of  sympathetic  attraction,  into  one  finely- 
constituted  whole,  so  that  not  one  heart  can  throb  but  all  hearts 
must  throb  with  it.  "  There  is  joy  in  the  presence  of  the  angels 
of  God  over  one  sinner  that  repenteth ;  "  and  to  every  man  who 
falls  into  sin,  we  may  say,  with  deeper  meaning  than  that  of  the 


THE    SPHERE    OF   HUMAN    INFLUENCE.  53 

prophet,  "  Hell  from  beneath  is  moved  for  thee,  to  meet  thee  at 
thy  coming." 

As  by  the  law  of  gravity  the  material  universe,  and  by  that 
of  love  the  spiritual  world,  so  by  the  association  of  ideas  the 
world  of  thought  is  bound  into  one  whole,  whereof  you  cannot  find 
one  thought  that  is  not  connected,  more  or  less  directly,  with  all. 
Nothing  known,  nothing  thought,  nothing  done,  nothing  felt,  fails 
to  leave  a  clew  by  which  it  may  be  recalled  to  memory.  Each 
moment's  state  of  consciousness  is  connected  in  a  train  which 
reaches  back  to  the  earliest  moments  of  life,  and  shall  reach  on 
unbroken  through  eternity,  so  that  it  must  ever  be  among  the  pos- 
sibilities of  memory  to  recall  the  thoughts  of  any  instant.  And  as 
the  rare  occurrence  of  unusual  power,  developed  by  accidental 
excitement,  suggests  hopes  of  an  indefinite  increase  of  power, 
when  we  shall  have  laid  aside  this  frame,  subject  to  accidents,  so 
the  preternatural  manifestation  of  memory,  in  certain  states  of 
health,  warns  us  that  this  possibility  of  recalling  all  things  may 
become  an  actual  reality  in  the  future  life. 

Then,  as  the  soul  surveys  the  past,  with  memory  presenting  its 
record  of  every  thought  and  word  and  deed,  and  with  an  eye 
quickened  to  see  the  influence  which  each  has  had,  she  may  sit  in 
judgment  on  her  own  character ;  and  the  word,  which  shall  be 
the  final  judge,  may  speak  through  the  soul  itself.  Then,  also,  as 
she  enters  the  company  of  cherubim  and  seraphim,  they  will  need 
no  record  of  her  good  or  evil  deeds  other  than  that  written  upon 
herself ;  from  their  eyes,  as  from  her  own,  there  shall,  when  she 
is  present,  be  no  past  sin  hidden,  and  no  good  thought  concealed. 
In  the  anticipation  of  standing  before  such  a  tribunal,  who  can  fail 
to  find  strength  to  resist  the  tempter,  and  encouragement  in  striv- 
ing after  good  ? 


XII. 

MAGNITUDE. 

IN  the  minds  of  some  who  have  read  certain  of  the  preceding 
chapters  there  has,  doubtless,  arisen  an  objection  to  accepting  the 
results  obtained,  —  the  objection  that  the  results  are  infinitesimal, 
and  therefore  non-existent.  De  minimis  non  curat  lex  ;  and  that, 
it  may  be  said,  will  be  the  opinion  of  the  court  of  conscience  con- 
cerning the  "  permanent  impression  of  our  words  upon  the  air  ;  " 
concerning  the  effect  of  our  motions  upon  the  distant  stars ;  con- 
cerning the  influence  of  our  character  upon  the  tone  of  the  spiritual 
universe.  These  influences  will  be  infinitesimal,  and  to  a  large 
part  will  balance  and  destroy  each  other. 

In  reply  to  these  objections,  I  will  begin  by  observing  that  the 
balancing  of  two  forces  is  a  real  effect  in  nature,  not  to  be  for  a 
moment  confounded  with  the  non-existence  or  destruction  of  the 
forces.  If  the  earth  do  not  rise  to  meet  this  falling  rain-drop,  it 
demonstrates  that  another  drop,  or  its  equivalent,  is  falling  on  the 
other  side  of  the  globe. 

Let  me  also  concede,  at  once,  that,  in  the  form  in  which  the 
arguments  have  been  put,  there  is  an  assumption  of  certain  laws 
of  physics,  which,  being  laws  deduced  from  observation,  may  be 
subject  to  perturbations  not  yet  discovered.  Thus  Babbage,  in  the 
chapter  alluded  to,  assumes  that  the  wave  of  sound  runs  friction - 
less  through  the  air,  the  heat  developed  by  the  compression  being 
absorbed  in  the  expansion.  But  the  experiments  of  Uriah  Boyden 

(54) 


MAGNITUDE.  55 

have  since  shown  that  there  is  a  slight  amount  of  heat  developed 
by  the  friction  of  the  wave  ;  and  this  would  slowly,  but  constantly, 
diminish  the  force  employed  in  propagating  the  sound  as  sound. 
Again,  I  have  assumed  that  gravity  is  a  force  acting  at  a  distance, 
and  requiring  no  time  whatever  for  its  transmission ;  but  it  may 
possibly  be  hereafter  shown  that  its  speed  is  not  thus  absolutely 
infinite. 

With  regard,  however,  to  the  main  objection,  that  the  infinitesi- 
mal may  be  neglected,  the  objection  appears  to  me  not  valid,  and 
to  arise  from  the  weakness  of  the  human  imagination.  "  Time  and 
space  are  great  only  with  reference  to  the  faculties  of  the  beings 
which  note  them."  In  space  and  time  themselves  there  is  no 
natural  unit,  or  scale  of  magnitude  ;  these  are  given  only  by 
thought,  manifesting  itself  through  material  phenomena.  Hence 
every  scale  of  magnitude  is  relative  to  the  mind  employing  it,  and 
it  is  only  the  Unlimited  Mind  that  can  be  free  from  the  fetters 
imposed  upon  thought  by  the  scale  employed. 

In  some  of  the  nebulae  we  have  examples  of  the  spira  mirabilis 
of  Bernoulli,  drawn  on  a  gigantic  scale,  so  that  tbe  part  visible  to 
the  telescope  is  many  millions  of  leagues  in  extent.  This  spiral 
may  be  drawn,  on  a  smaller  scale,  by  tracing  upon  a  circumpolar 
map  the  path  of  a  ship  keeping  steadily  upon  any  one  course,  not 
to  a  cardinal  point.  Let  us  imagine  this  map  extended  until  its 
radiating  meridians  stretch  out  among  the  stars,  and  let  us  trace 
upon  it  a  spira  mirabilis,  which,  beginning  at  a  distance  of  one 
billion  kilometers  from  the  pole,  is  running  thirty  degrees  north  of 
east.  The  length  of  this  spiral  will  be  two  billion  kilometers,  and 
it  will  make  an  innumerable  number  of  revolutions  in  reaching  the 
pole.  Let  us,  next,  imagine  that  we  have  come  in  upon  the  spiral, 
until  we  are  but  one  kilometer  from  the  pole  ;  the  length  of  spiral 


56  GEOMETRY   AND   FAITH. 

yet  remaining  will  be  two  kilometers,  and  the  number  of  revolu- 
tions still  to  be  made  around  the  pole  will  remain  innumerable. 
Let  us  again  approach  until  we  are  within  one  millimeter  from  the 
pole  ;  the  remaining  length  of  spiral  will  be  two  millimeters  ;  but 
the  number  of  revolutions  yet  to  be  made  about  the  pole  will  still 
be  innumerable.  This  little  central  part  of  the  spiral,  all  lying 
within  a  circle  two  millimeters  in  diameter,  will  be  precisely  simi- 
lar in  shape  to  the  whole  spiral,  two  billion  kilometers  in  diameter. 
If  it  were  possible  to  look  at  this  central  part  with  a  lens  that 
should  magnify  a  quadrillion  times,  it  would  appear  precisely  of 
the  same  size  and  shape  as  the  whole  spiral.  By  your  approach  to 
the  centre,  the  scale  alone  would  be  altered  ;  the  central  part  of 
the  spiral  would  be  a  reduced  picture  of  the  whole  ;  its  linear 
dimensions  would  be  the  fifteenth  place  of  decimals  of  the  dimen- 
sions of  the  original  spiral. 

That  is,  it  is  the  nature  of  the  spira  mirabilis,  that  whether  you 
move  inward  toward  the  pole,  or  outward  away  from  it,  the  part 
between  your  position  and  the  centre  remains  always  exactly  of 
the  same  shape,  differing  only  in  scale.  Let  us  then  approach  to 
within  the  hundredth  of  a  millimeter  from  the  pole  ;  the  central 
portion  becomes  now  too  small  to  be  seen  distinctly  by  the  naked 
eye ;  but  it  is  of  same  shape  as  the  whole ;  and  although  its  total 
length  is  only  the  fiftieth  of  a  millimeter,  it  still  makes  an  infinite 
number  of  convolutions  about  the  pole.  On  the  other  hand,  if  we 
should  run  out  to  the  distance  of  a  trillion  kilometers,  we  should, 
probably,  reach  the  distance  of  the  furthest  star  visible  to  the 
naked  eye.  That  is  to  say,  the  unaided  eye  enjoys  a  range  of 
vision  through  about  twenty  places  of  decimals  in  linear  dimension. 
And  if  we  should  revise  Archimedes'  calculation,  on  the  number 
of  sand-grains  requisite  to  bury  the  universe,  we  should,  conse- 


MAGNITUDE.  57 

quently,  see  that  sixty  places  of  figures  will  express  the  number  of 
grains  of  the  finest  silt  requisite  to  fill  all  space,  out  to  the  stars 
of  the  sixth  magnitude.  The  sixtieth  place  of  decimals  is,  there- 
fore, not  zero  ;  it  is  the  ratio  of  the  space  occupied  by  a  minute 
grain  of  fine  sand,  to  the  space  in  which  the  fixed  stars  lie.  In 
that  minutest  space  the  same  forms  may  lie  concealed  as  those 
which  are  illustrated  in  the  whirlpool  nebulae,  and  which  might  be 
conceived  as  filling  vaster  spaces.  Twenty  places  of  figures,  in 
linear  dimension,  carry  us  beyond  the  limits  of  sight ;  but  space 
does  not  end  there  ;  and  we  might  affix  figures  forever,  without 
arriving  at  a  magnitude  so  great  as  to  be  impossible. 

Turning  our  thoughts  again  to  the  minute  central  portion  of  the 
spira  mirabilis,  we  could,  by  the  aid  of  the  microscope,  see  an 
object  whose  linear  dimensions  would  stand  in  the  seventh  or 
eighth  place  of  decimals  of  a  meter.  In  that  central  speck,  visible 
only  to  the  best  microscope,  the  wonderful  spiral  would  still  exist 
in  all  its  perfection  ;  still  making  its  proper  angle  with  a  line  to 
the  absolute  centre  ;  still  being  in  its  fixed  proportion  of  length 
to  the  length  of  that  line  ;  and  still  making  an  infinite  number  of 
convolutions  around  the  pole.  And  this  would  hold  could  we,  by 
the  aid  of  more  and  more  powerful  lenses,  continually  approach 
the  centre  until  our  distance  from  it  stood  in  the  ninth  or  ninetieth, 
the  nine  hundreth  or  the  nine  millionth,  place  of  decimals.  We 
may  write  cyphers,  after  the  decimal  point,  at  the  rate  of  nine 
million  a  second,  for  nine  million  centuries,  but  when  we  finally 
write  a  significant  figure,  that  figure  is  not  a  cypher ;  it  is  signifi- 
cant ;  and  if  it  signifies  the  distance  which  yet  remains  between 
our  moving  point  and  the  pole  which  it  is  approaching,  then,  incon- 
ceivably small  as  the  distance  is,  it  has  its  relations :  it  is  one-half 
the  length  of  the  remaining  portion  of  the  spiral ;  and  that  portion 


58  GEOMETRY  AND   FAITH. 

of  the  spiral,  nearly  as  it  may  be  without  any  length  or  size,  still 
makes  its  infinite  number  of  convolutions  about  the  pole. 

It  has  been  inferred  that,  because  the  scale  is  thus  capable  of 
indefinite  expansion  and  contraction,  without  any  destruction  of  the 
form,  or  law  of  the  curve,  the  contraction  might  proceed  to  reduce 
the  spiral  into  the  absolute  point  at  the  centre  ;  and  that  afterward 
the  point  might  be  considered  as  an  absolute  nonentity,  without 
destroying  the  spiral.  Thus  space  and  time,  it  has  been  said,  may 
be  shown  to  be  purely  subjective. 

But  the  failure  in  this  attempt  to  demonstrate  the  subjective 
character  of  space  is  twofold.  A  point  is  not  a  nonentity  ;  it  is  a 
zero  of  magnitude  ;  yet  it  has  position,  or  is  a  position.  It  is  not 
in  the  mind,  it  is  in  space,  fixed  in  its  position,  although  without 
magnitude.  But  although  thus  real,  and  indestructible  in  space, 
yet  being  without  dimensions  or  parts,  it  is  incapable  of  containing 
a  curve  -except  by  a  figure  of  speech  ;  by  which  we  either  attribute 
to  the  point  the  potentiality  of  the  subjective  law ;  or  else  speak 
of  a  point  when  we  simply  mean  an  infinitesimal  space.  As  we 
diminish,  for  example,  the  scale  of  the  spira  mirabilis,  —  by  run- 
ning in  upon  it  towards  the  centre,  and  considering  only  the  part 
yet  remaining,  which  is  always  similar  to  the  original  whole,  — 
it  remains  real,  so  long  as  we  have  not  arrived  at  the  actual 
centre  ;  but,  when  that  point  is  reached,  the  spiral  has  vanished  ; 
there  is  nothing  remaining  between  us  and  the  pole,  for  we  are  at 
the  pole.  That  pole  does  not  then  become  subjective  ;  it  is  a  real 
point ;  but  it  does  not  contain  the  spiral,  except  potentially. 

In  dealing  with  the  infinitesimal  and  the  infinite,  the  practice 
of  geometers  varies ;  some  delight  in  stating  truths  in  forms  which 
seem  false  and  self-contradictory  ;  others  carefully  avoid  such 
forms.  Both  classes  are  liable  to  error ;  the  impossibility  of 


MAGNITUDE.  59 

clearly  imagining  the  infinite  and  the  infinitesimal  is  not  destroyed 
by  any  forms  of  language  ;  and  the  difficulty  of  reaching  true  con- 
clusions is  not  necessarily  increased  by  the  use  of  forms  of  speech 
concerning  them,  literally  incorrect.  The  calculus  of  Leibnitz  is, 
in  a  majority  of  its  propositions,  literally  false  ;  yet  it  always  leads 
to  true  conclusions,  in  the  hands  of  one  who  can  distinguish 
between  the  letter  and  the  spirit. 

The  best  geometer  can,  however,  err ;  as,  for  example,  some 
have  said  that  the  point  approaching  the  pole,  in  a  spira  mirabilis, 
can  never  reach  the  pole,  because  it  will  always  have  an  infinite 
number  of  revolutions  to  make  before  reaching  it.  The  premise 
is  true,  but  the  conclusion  false.  The  point  will  always,  until  it 
reaches  the  pole,  have  an  infinite  number  of  revolutions  to  make, 
before  reaching  it.  But  if  the  motion  in  the  spiral  be  with  uniform 
velocity,  then  the  revolutions  will  be  with  increasing  velocity  ;  and 
finally  with  infinite  velocity  ;  so  that  the  infinite  number  of  revolu- 
tions will  be  accomplished  in  a  finite  time.  To  take  the  particular 
example  we  have  been  considering,  —  the  point  moving  at  a  con- 
stant angle  of  sixty  degrees  with  the  line  to  the  pole,  —  if  the 
point  moves  uniformly  at  the  rate  of  a  slow  walk,  say  four  kilo- 
meters an  hour,  then  it  will  reach  the  pole  in  precisely  one  hour 
from  the  time  that  its  polar  distance  is  two  kilometers.  But,  while 
the  motion  of  the  point  is  uniform,  at  four  kilometers  an  hour,  and 
its  approach  to  the  pole  uniform,  at  half  that  rate,  the  angular 
velocity  constantly  increases,  being  always  such  that  (if  it  could 
remain  uniform)  it  would  carry  the  point  around  the  pole  in  about 
three  and  five-eighths  the  time  that  remains  of  the  hour.  Thus, 
when  at  the  distance  of  a  meter  from  the  pole,  the  revolution 
would  carry  it  around  nine  times  a  minute  ;  while  at  the  distance 
of  a  millimeter  the  revolution  would  be  at  the  rate  of  one  hundred 


60  GEOMETRY   AND    FAITH. 

and  fifty  circuits  a  second  ;  the  rotation  thus  increasing  in  rapidity, 
in  direct  proportion  to  the  approach  to  the  pole.  But  in  less  than 
the  one  five-hundredth  of  a  second  the  hour  has  expired ;  the  centre 
is  reached  ;  the  whole  spiral  has  been  passed  over,  and  the  point, 
having  made  innumerable  revolutions,  is  at  the  pole.  The  pole 
does  not  contain  the  spiral ;  although  it  may  be  said  to  contain  it, 
in  the  sense  that  the  spiral,  with  an  infinite  number  of  coils,  is  con- 
tained in  any  portion  of  space  around  the  pole,  however  small,  even 
if  smaller  than  any  portion  that  can  be  measured,  named,  or  im- 
agined. Thus  the  point  may  be  said  to  contain  the  spiral  poten- 
tially, so  that  the  spiral  would  become  actual,  could  the  point  be 
magnified. 

This  illustration  of  the  logarithmic  spiral  has  been  chosen  and 
fully  expanded,  not  only  because  of  its  peculiar  adaptation  to  the 
illustration  of  similarity,  and  its  curious  property  of  always  having, 
even  when  reduced  to  an  infinitesimal  size,  an  infinite  number  of 
coils,  but  for  the  historic  associations  with  that  curve  which  Ber- 
nouilli  would  fain  have  had  carved  upon  his  tombstone.  The  same 
conclusions  would  be  reached  did  we  consider  the  shrinking  of  any 
other  forms  in  just  proportion.  It  is  conceivable  that  the  entire 
universe  might  be  altered  in  size,  and  if  proportional  changes  went 
on,  in  every  part,  in  all  the  forces  acting  upon  it,  and  in  the  facul- 
ties of  all  creatures,  it  would  not  be  in  the  power  of  human  beings 
to  discover  that  the  change  was  made.  Were  the  universe  thus 
reduced  to  the  twentieth  place  of  decimals  in  linear  dimensions, 
and  the  requisite  changes  made  in  the  forces  of  nature  and  in  the 
faculties  of  man,  the  whole  stellar  system  would  be  contained  in 
the  space  now  occupied  by  a  grain  of  sand.  A  second  reduction 
might  take  place,  and  a  third,  and  so  on  forever ;  and,  unless  the 
rate  at  which  the  changes  were  made  increased,  the  universe  would 


MAGNITUDE.  61 

still  remain  to  its  inhabitants  as  large  and  grand  as  before.  But 
let  the  rate  increase  so  as  to  bring  the  universe  to  a  point,  and  its 
actuality  would  be  gone,  and  its  potentiality  alone  remain.  Still 
that  potentiality  would  be  objective  to  the  Creative  Mind  ;  objec- 
tive in  the  point.  Sweep  the  point  away,  and  the  universe  would 
exist  only,  as  in  Erigena's  "  Second  Division  of  Nature,"  subjec- 
tively in  the  Divine  Thought. 

We  thus  reach  sublimer  conceptions  of  the  immeasurable  gulf 
between  the  human  arid  the  Divine  Mind  by  holding  to  the 
veracity  of  that  intuition  which  pronounces  space  and  time  inde- 
structible objective  entities  ;  and  renouncing  the  philosophic  wis- 
dom, made  popular  by  Teufelsdreck,  which  accounts  them  mere 
modes  of  human  thought.  To  the  human  mind  there  is  no  unit 
of  space  or  of  time,  save  those  given  in  the  creation  ;  we  cannot 
even  imagine  any  unit  not  thus  given.  To  the  Divine  Mind  alone 
belongs  the  possibility  of  deciding  on  the  scale  of  creation,  and 
deciding  what  men  shall  consider  large  or  small,  brief  or  lasting. 
As  the  human  eye  requires  the  aid  of  the  telescope  at  one  end  of 
its  range  of  vision,  and  the  microscope  at  the  other,  so  all  our  fac- 
ulties are  limited  to  that  which  is  neither  too  great  nor  too  small 
for  us.  But,  in  regard  to  the  works  of  nature,  we  neither  discover 
limits,  nor  are  we  compelled  by  any  mental  necessity  to  suppose 
that  there  are  limits.  The  bounds  of  the  universe  are  independent 
of  the  weakness  and  limitations  of  our  powers  of  imagination. 

There  is,  therefore,  no  impossibility  in  the  speculations  of  Lov- 
ering,  published  in  the  "  Cambridge  Miscellany  "  in  1842;  that 
the  atoms  of  our  universe  may  be  stars  and  suns  of  a  smaller  one, 
composed  in  like  manner  of  infinitely  smaller  stars  and  suns  ;  while 
our  constellations  and  solar  systems  may  constitute  only  molecules 
in  a  vaster  world.  The  infinity  of  space  would  almost  seem  to 


62  GEOMETRY   AND   FAITH. 

demand  such  an  arrangement  to  utilize  its  wastes.  The  human 
mind,  fettered  by  the  body,  seems  in  such  speculations  to  show  its 
kindred  to  the  Infinite  Spirit,  to  whom 

"  There's  nothing  great  appears, 
.  .  .  There's  nothing  small." 

And  these  speculations  are  not  confined  to  a  few  learned  men, 
whose  studies  lead  them  naturally  to  the  theme.  Fifteen  years 
before  the  publication  of  Lovering's  paper,  I  myself,  a  child,  heard 
other  children  supposing  that  this  universe  might  be  the  atoms  in  a 
crumb  let  fall  by  a  giant ;  and  that  the  slow  precession  of  the 
equinoxes  might  be  the  rotation  of  that  crumb,  in  the  air,  as  it  fell 
to  the  ground,  in  an  infinitely  larger  universe. 

The  true  greatness  of  a  work  is  in  the  thought  which  it  embodies, 
not  in  the  scale  on  which  it  is  wrought.  The  idea  or  law  of  the 
spira  mirdbilis  has  a  fascinating  beauty  to  a  geometer  ;  but  to  such 
a  mind  it  is  a  matter  of  perfect  indifference  whether  that  spiral  be 
illustrated  in  the  minutest  shell  or  in  the  largest  nebula.  I  have 
been  sometimes  as  much  moved  at  a  slight  wood-cut  outline  of  a 
mountain  range  as  at  the  sight  of  the  vast  masses  themselves.  The 
Dead  March  from  Saul  will  express  grief  on  a  grand  scale,  a  sense 
of  human  weakness  resting  in  unshaken  confidence  on  the  Divine 
strength,  whether  played  on  a  single  instrument  or  with  a  full 
orchestra. 

Our  speculations  on  the  scale  of  magnitude  become,  therefore, 
unimportant.  Whether  there  are,  or  are  not,  infinitesimal  worlds 
included  in  the  atoms  of  our  worlds,  and  infinite  worlds  in  whicL 
our  systems  are  atoms,  the  grandest  and  most  inspiring  object  for 
our  contemplation  is  the  law,  plan,  or  thought,  on  which  the  uni- 
verse, within  reach  of  our  faculties,  is  built.  We  need  not  reduce 


MAGNITUDE.  63 

it  to  a  point,  or  dissolve  it  in  ideal  subjectivity,  in  order  to  show 
its  unity.  The  universality  of  gravity,  the  co-extensive  universality 
of  light,  which  by  the  spectroscope  has  shown  that  some,  at  least? 
of  our  chemical  elements  are  universally  diffused,  the  correlation 
of  forces,  the  adaptation  of  all  parts  of  the  universe  to  each  other, 
all  tend  to  confirm  the  sublime  conclusion  that  the  universe  is  the 
expression  of  one  infinitely  complex,  yet  infinitely  simple,  connected 
thought,  in  which  all  was  foreseen,  and  all  comprehended  at  a  sin- 
gle glance  by  the  Intelligence  which  framed  it.  The  aim  of  science 
is  to  develop  and  trace  the  connection  of  the  parts  of  this  intel- 
lectual whole  ;  the  end  of  religion  is  to  interpret  for  the  heart 
and  soul  the  lessons  given  through  this  intellectual  form. 


XIII. 

CHANCE  AND  AVERAGE. 

WHEN  two  phenomena  arise  from  entirely  independent  causes, 
the  relation  of  one  to  the  other  is  said  to  result  from  chance.  The 
disposition  to  consider  chance  an  actually  existing  cause  is  so 
great,  that  men  have,  in  all  ages,  personified,  and  in  some  nations 
even  deified  it. 

In  the  highest  contemplation  of  the  universe,  as  the  realization 
of  one  grand  conception  of  the  Divine  Mind,  it  might  be  thought 
that  the  idea  of  chance  would  be  excluded,  because  all  phenomena 
would  then  be  regarded  as  springing  from  a  single  cause  ;  all  the 
minutest  events  would  be  considered  not  only  as  foreseen,  but  as 
intended  ;  as  the  necessary  results  of  the  original  thought  made 
actual  in  the  universe. 

But  the  idea  of  chance,  of  relations  in  events  springing  from 
independent  causes,  is  so  positive  in  its  character,  that  we  are 
unwilling  to  concede  it  to  be  a  mere  result  of  the  weakness  of  the 
human  mind,  of  our  inability  to  rise  to  a  habitual  contemplation  of 
one  First  Cause.  It  seems  more  like  a  direct  gift  of  power ;  a 
power  to  apprehend  some  really  occurring  phenomenon  in  nature. 
As  such,  it  forms  the  basis  of  a  distinct  and  valuable  calculus, 
applicable  to  important  economic  questions  of  assurance  and  annui- 
ties, and  to  weighty  scientific  problems,  as  a  test  of  hypotheses, 
and  a  criterion  for  rejecting  doubtful  observations.  The  successful 
application  of  this  calculus  of  probabilities  to  so  many  actual  prob- 

(64) 


CHANCE    AND    AVERAGE.  65 

lems  in  the  universe  is  a  demonstration  that,  however  difficult  it 
may  be  to  reconcile  the  conception  with  our  ideas  of  ' '  foreknowl- 
edge absolute  "  in  the  single  Creative  Will,  we  must,  nevertheless, 
admit  into  our  theory  of  the  world  the  conception  of  independent 
causes,  leading  to  what  may  be  justly  called  accidental  results. 

The  reconciliation  of  this  contradiction,  so  far  as  reconciliation  is 
possible  in  our  finite  minds,  is  probably  to  be  found  in  the  con- 
sideration of  averages.  In  our  human  work  we  frequently  act 
upon  a  multitude  of  individual  objects,  without  special  designs  in 
regard  to  each,  but  with  a  general  regard  to  the  average  action 
and  to  the  total  result.  The  sower  does  not  consciously  choose 
the  position  in  which  a  single  grain  of  his  wheat  shall  fall,  yet 
designs  and  accomplishes  an  even  cast  of  a  given  quantity  of  seed 
to  the  acre.  The  causes  which  determine  the  position  of  each 
grain  are  so  numerous,  and  their  connection  so  remote,  that  they 
may  be  considered,  for  one  grain,  independent  of  those  for  another. 
In  throwing  a  die  repeatedly,  in  like  manner,  the  causes  which 
determine  its  position  after  one  throw  are  so  numerous,  and  so  re- 
motely connected  with  those  that  determine  its  next  position,  that 
they  may  be  considered  independent.  Yet  the  throws  are  so 
governed  by  our  will  that  we  may  decide,  beforehand,  how  many 
to  make  in  each  minute,  —  and  the  positions  are  so  determined  by 
the  shape  and  material  of  the  die,  that  if  it  be  a  homogeneous 
cube  the  tendency  will  be,  as  the  throws  are  multiplied,  to  have 
each  side  uppermost  one-sixth  of  the  time. 

This  is  the  law  of  chance,  as  applied  to  averages.  And  as 
chance  has  been  personified,  and  even  deified,  so  average  has,  by 
some  writers,  had  divine  powers  ascribed  to  it.  It  has  been  gravely 
asserted  that  the  saving  of  a  man  from  criminal  courses  only  drives 
another  man  into  crime  to  keep  up  the  average ;  as  though  the 


66  GEOMETRY   AND    FAITH. 

present  average  had  an  inherent  power  to  perpetuate  itself;  as 
though  dice  could  not  be  loaded  without  producing  a  counter- 
loading  in  the  other  player's  dice  ;  as  though  the  sower  could  not 
vary  the  size  or  cast  of  one  handful  without  immediately  varying 
another  handful  to  keep  his  field  from  having  more  or  less  seed 
upon  its  surface. 

The  average  is  a  result,  not  a  cause.  It  is  the  result  of  rela- 
tions that  exist  between  the  'various  causes  producing  the  effects, 
and  may  be  changed  at  any  time  by  interfering  between  those 
relations ;  by  the  dice  being  loaded,  or  the  sower  walking  at  a  dif- 
ferent pace  ;  by  Jenner's  introduction  of  vaccination,  or  the  dis- 
covery of  America  putting  quinine  into  the  physician's  hand ;  or 
John  Howard  visiting  the  prisons,  or  the  Apostle  Paul  receiving 
his  commission.  Great  changes  thus  take  place  in  natural  aver- 
ages ;  and  small  changes  may  be  made  at  any  moment  by  the 
action  of  the  human  will. 

It  seems  not  unworthy  our  highest  conceptions  of  the  divine  plan 
to  suppose  that  certain  groups  of  phenomena  in  nature  may,  like 
the  sowing  of  seed  by  man,  be  directed  and  intended  for  average 
results,  without  special  design  for  each  individual  case.  The  im- 
passable gulf  between  the  finite  and  the  Infinite  Mind  would  still 
remain,  in  the  ability  of  the  Divine  Providence  to  select  at  will 
any  one  of  the  innumerable  cases,  and  employ  it  as  a  means  to 
higher  and  further  ends.  The  winds  from  the  Mediterranean,  for 
example,  bring  a  fixed  average  of  vapor  to  the  summits  of  the 
Alps,  where  it  is  showered  down  in  countless  crystals  of  snow. 
These,  under  slight  changes  of  temperature,  contract  into  minute 
globes  ;  and  these  particles  of  ice,  piled  up  in  the  mountain  basins, 
press  themselves  into  an  almost  solid  mass,  and  push  themselves, 
or  their  earlier  companions,  down  the  valleys,  grinding  off  the 


CHANCE    AND    AVERAGE.  67 

rocks  into  powder,  which  is  washed  by  the  melting  ice  into  the 
rivers  and  into  the  sea.  The  magnitude  of  these  glaciers  is  limited 
by  the  quantity  of  snow ;  and,  in  its  turn,  limits  the  quantity  of 
gravel  and  sand,  and  the  size  of  the  boulders  formed  by  them. 
To  these  results,  a  thousand  causes  which  I  have  not  mentioned 
conspire.  A  theist,  believing  that  the  glacial  system  of  the  mod- 
ern Alps  is  part  of  a  divine  plan,  is  not  obliged  to  suppose  that 
this  plan  includes  the  position  of  every  individual  grain  of  silt,  in 
the  ocean  bed,  brought  from  the  Alps ;  is  not  obliged  to  make  this 
supposition,  even  if  to  his  theism  he  adds  faith  in  a  special  Provi- 
dence, and  thinks  that  a  grain  of  sand  may  be  a  providential 
instrument  in  effecting  some  great  result. 

But  whatever  our  explanation  of  the  occurrence  of  so-called 
chance  among  the  averages  of  nature,  these  chances  and  averages 
are  frequently  adapted  to  each  other  with  a  harmony  that  seems 
to  admit  of  no  other  solution  than  a  reference  to  the  Divine  plan 
which  fits  each  to  all  and  all  to  each.  Enthusiastic  students  of  the 
calculus  of  probabilities  sometimes  represent  all  human  judgments 
as  the  result  of  a  calculation  of  chances  ;  and  our  certainties  are 
said  by  them  to  be  merely  propositions,  the  truth  of  which  is  in- 
finitely probable.  Many  of  the  arguments  of  natural  theology,  so 
called,  can  be  very  conveniently  put  into  this  form.  In  the  forma- 
tion of  planets  around  the  sun,  according  to  the  nebular  hypothesis, 
the  chances  were  small  against  an  order  which  should  fail  to  pre- 
serve the  stability  of  the  system  ;  and  the  present  harmony  of  dis- 
tances must  be  referred,  directly  or  indirectly,  to  presiding  thought. 
In  the  formation  of  the  solar  system,  the  chances  were  small  that 
this  particular  planet  should  have  its  elements  mingled  in  precisely 
that  proportion  which  has  resulted  in  so  full  a  development  of  life 
and  of  human  activity ;  and  the  arguments  of  Prof.  Cooke's  "  Re« 


68  GEOMETRY  AND    FAITH. 

ligion  and  Chemistry"  derive  from  this  consideration  a  demon- 
strative force. 

In  the  course  of  this  successive  development  of  vegetable  and 
animal  life  upon  the  earth,  there  has  been,  with  frequent  muta- 
tion, a  general  permanence.  Scientific  speculation  at  the  present 
time  busies  itself  with  the  question  whether  the  permanence  has 
been  real,  and  the  changes  sudden  ;  or  whether  the  stability  is 
seeming,  and  the  mutations  have  always  been  going  on  with 
stealthy  step.  Whichever  of  these  theories  proves  to  be  true, 
that  of  Plato,  or  that  of  Democritus,  there  is  a  seeming  stability 
in  the  present  species,  which  have  lasted  without  sensible  change, 
except  the  extinction  of  some  kinds,  for  thousands  of  years.  Ac- 
cording to  the  theory  of  Democritus,  as  revived  in  our  days,  this 
arises  from  the  fact  that  the  species  at  present  existing  are  the 
fittest  for  the  existing  epoch,  and  thus  survive.  According  to  the 
rival  theory  this  fitness  arose  from  no  blind  struggle  for  life,  but  in 
accordance  with  a  Divine  plan,  fulfilled  by  divine  power.  In  ordi- 
nary cases  the  judgment  may  possibly  remain  suspended,  whether 
to  suppose  the  Divine  Will  acted  in  reference  to  the  perpetuation 
of  a  species  by  some  general  law,  covering  many  species,  or  by 
special  adaptations  to  one.  These  cases  may  therefore  be  dis- 
missed from  the  argument.  If  we  grant  that  a  blind  evolution  by 
natural  variation  and  survival  of  the  fittest  will  explain  them,  it 
must  also  be  conceded  that  an  intelligent  adaptation  of  the  organ- 
ism to  its  medium  will  also  explain  them.  But  there  are  other 
cases,  in  which  the  imagination  runs  riot  in  vain  for  any  "  suffi- 
cient reason ' '  outside  of  the  will  and  purpose  of  the  Creator,  ful- 
filling an  original  plan.  These  cases  are  like  "  experimenta  cru- 
cis,"  —  the  theory  that  fails  to  hint  at  a  possible  explanation  fails 
to  explain  the  universe. 


CHANCE    AND    AVERAGE.  69 

Such,  it  appears  to  me,  are  the  cases  in  which  the  fecundity  of 
a  creature  is  in  inverse  proportion  to  its  chances  of  life.  I  would 
by  no  means  say  that  these  are  the  only  points  in  the  animal  and 
vegetable  economy  which  the  evolution  theories  appear  to  me  to  be 
utterly  incapable  of  explaining  ;  but  they  are  the  cases  which  fall 
under  the  head  of  average  and  chances,  and  demonstrate  that  the 
Eternal  Thought  which  planned  this  present  world  comprehended 
all  and  more  than  all  which  is  included  in  our  modern  calculus  of 
probabilities. 

If  the  ovaries  of  the  Dodo  contained  one  thousand  ova,  and  if, 
on  an  average,  less  than  one  of  these  grew  into  an  adult  female 
Dodo,  with  equal  chances  of  propagating  her  kind,  it  is  evident 
that  the  Dodo  must  become  extinct.  If,  on  the  other  hand,  two 
of  these  ova  were  impregnated,  and  came  to  maturity,  it  would 
take  but  a  few  generations  of  the  bird  to  cover  the  earth,  and 
exclude  all  other  beings.  This  is  prevented,  it  may  be  said,  by 
the  struggle  for  life.  But  the  fecundity  of  each  species  must  be 
exactly  proportioned  to  the  chances  of  failure  in  that  struggle. 

The  horse-hair  eel  is  said  to  lay  several  millions  of  e^gs  ;  let  us 
say  five  million.  Why  this  enormous  fecundity  ?  Because  the 
chances  of  the  eggs  coming  to  maturity,  as  eels,  is  so  small.  In 
order  to  keep  the  species  in  existence,  two  in  five  millions  (if  the 
sexes  are  of  equal  numbers)  must  succeed  in  escaping  all  the 
dangers  which  beset  the  eggs  and  the  young  in  the  brook,  and 
then  succeed  in  finding,  near  the  brook,  crickets  or  grasshoppers 
into  which  they  may  penetrate.  These  grasshoppers  must  escape 
their  enemies,  and  survive  the  depredations  of  the  hair  eels,  until 
the  latter  reach  maturity,  when  they  must  escape  near  enough  to 
a  brook  to  find  their  way  there,  and  meet  hair  eels  of  the  opposite 
sex.  The  chances  are  two  in  five  millions,  let  us  say,  and  the 


70  GEOMETRY   AND    FAITH. 

creature  lays  five  millions  of  eggs.  Did  she  average  but  four 
millions,  the  race  would  in  a  few  years  become  extinct ;  did  she 
average  six,  the  creatures  would  multiply  in  a  few  years  beyond 
all  bounds.  The  permanence  of  the  species  for  so  many  years 
demonstrates  the  accuracy  with  which  its  fecundity  is  proportioned 
to  the  slimness  of  its  chance  in  the  falsely-called  struggle  for  life. 


XIV. 

PHYLLOTAXIS. 

THE  reader  may  be  interested  in  a  more  detailed  development 
of  the  arguments  briefly  alluded  to  upon  pages  5  and  6,  35  and  36. 

There  is  a  rule  in  arithmetic  called  the  rule  of  False,  or  the 
rule  of  Position.  It  is  the  most  general,  and  the  most  useful  of 
all  rules  in  the  art  of  computation.  Its  method  renders  it  appli- 
cable to  every  problem  in  which  the  accuracy  of  an  answer  can 
be  tested.  The  rule  may  be  briefly  stated  as  follows :  Guess  at  an 
answer,  and  test  by  numerical  computation  the  accuracy  of  its 
results.  If  the  results  are  accurate,  the  answer  stands  the  test 
and  was  correct.  And,  if  the  results  are  not  accurate,  the  error 
affords  data  for  estimating  the  error  of  the  assumed  answer ;  and 
making  a  second  better  guess,  to  be  tested  as  the  first  was ;  and 
thus  to  afford  data  for  a  third  guess,  still  nearer  the  truth. 

This  method  of  hypothesis  and  verification  is  applicable  not 
only  to  arithmetical  problems  but  to  all  questions  of  practical 
science.  It  may  even  be  said  to  be  the  general  rule  by  which  the 
finite  mind  approaches  every  truth  which  it  can  approach ;  and 
thus  reaches  every  truth  which  it  can  reach.  In  some  minds  the 
process  is  rapid  and  with  a  very  obscure  and  evanescent  con- 
sciousness of  its  operation ;  in  others  slower  and  with  distinct 
knowledge  of  the  steps ;  and  in  all  minds  the  rapidity  and  ease 
of  the  process  vary  with  the  nature  of  the  problem.  But  in 
regard  to  all  phenomena,  which  require  a  cause  or  theory  to 
account  for  them,  we  are  not  able  to  draw  the  theory  out  of  the 
facts ;  the  theory  comes  from  our  own  minds ;  we  place  it  among 

71 


72  GEOMETRY  AND  FAITH. 

the  facts;  and  then  adopt,  modify  or  reject  it,  according  to  its 
agreement  with  the  facts  and  ability  to  explain  them.  Some- 
times the  theory  explains  the  facts  perfectly,  but  goes  no  further; 
as,  for  example,  when  we  assume  a  centre  for  a  circumference 
passing  through  three  points ;  the  final  verification  of  the  centre, 
by  proving  it  equidistant  from  the  points,  gives  us  nothing  fur- 
ther to  do ;  we  have  found  the  centre,  that  is  all.  But,  in  more 
complex  cases,  it  frequently  happens  that  the  theory  proposed 
and  tested,  for  the  explanation  of  one  set  of  facts,  proves  to  be 
also  a  satisfactory  explanation  of  other  facts;  its  value  and 
authority  as  a  theory  are  thereby  immediately  greatly  enhanced. 
Nor  is  it  in  the  physical  sciences  alone  that  this  increase  of  cer- 
tainty and  value  may  be  found ;  the  operations  of  the  intellect  are 
similar,  on  whatever  class  of  objects  it  is  operating ;  and  theology 
itself  may  be  lawfully  treated,  to  a  certain  extent,  as  Spinoza 
treated  it,  by  mathematical  methods. 

Thus  Maupertuis  assumed  it  as  a  fundamental  principle  that 
the  Divine  Being,  being  unerringly  wise,  would  waste  no  energy ; 
that  everything  in  nature  must  therefore  be  done  with  perfect 
economy  of  force.  This  theological  dogma  is  called,  in  mechanics, 
the  principle  of  the  least  action ;  and  is,  in  that  science,  an  invalu- 
able and  fruitful  principle;  as  may  be  readily  shown  even  to 
those  not  skilled  in  mathematical  analysis.  Take  for  example  the 
mechanical  theory  of  light.  How  shall  light  move  in  a  uniform 
medium,  according  to  this  principle  of  the  least  action?  Evi- 
dently in  a  straight  line,  since  that  is  the  shortest  distance  between 
two  points.  How  shall  light  be  reflected  from  a  polished  surface  ? 
Evidently  so  as  to  make  the  sum  of  the  incident  and  reflected 
rays  the  shortest  possible  in  going  from  the  point  giving  light  to 
the  point  on  which  the  reflected  ray  shines;  and  a  simple  calcu- 
lation shows  that  this  requires  the  incident  and  reflected  ray  to 
make  the  same  angle  with  the  reflecting  surface.  How  shall  light 


PHYLLOTAXIS.  73 

be  refracted  in  passing  from  one  medium  to  another?  Here, 
again,  we  must  have  the  sum  of  the  incident  and  refracted  ray 
such  as  to  make  the  whole  power  expended  as  small  as  possible ; 
and  a  simple  calculation  proves  that  this  requires  the  sines  of  the 
angles  of  incidence  and  reflection  to  be  in  proportion  of  the  dif- 
ficulty of  moving  in  the  two  media.  These  instances  show  the 
usefulness  of  the  principle  of  least  action  in  physical  inquiry.  These 
results  concerning  light  are  amply  confirmed  by  experiment ;  and 
shown  to  give  the  manner  in  which  a  ray  of  light  actually  behaves. 
But  when  we  turn  the  argument  about,  and  from  this  agree- 
ment of  the  experimental  results  of  observations  on  light,  with 
the  principle  of  the  least  action,  would  argue  that  the  creator  of 
light  uses  a  perfect  economy  of  force,  and  is  therefore  unerring 
in  wisdom ;  we  feel  at  once  that  the  argument  is  not  absolutely 
conclusive.  It  creates  a  presumption,  but  does  not  force  a  con- 
viction. We  see  that  the  motion  of  light  in  a  straight  line,  reflec- 
tion at  an  equal  angle,  and  refraction  by  Snell's  law  of  sines, 
involve  perfect  economy  of  force ;  and  yet  these  may  be  neces- 
sary results  of  the  constitution  of  the  luminiferous  ether,  not 
foreseen  when  the  ether  was  constituted.  In  order  to  make  the 
argument  from  morphology  or  from  teleology  conclusive,  the 
instances  of  the  divine  thought  and  forethought  must  be  such  as 
cannot  be  explained  from  mechanical  or  mathematical  necessities. 
The  fact  that  Maupertuis's  theological  doctrine  of  the  divine 
economy  of  force  has  led,  or  could  have  led,  to  such  numerous 
discoveries  of  the  actual  laws  of  physics,  certainly  creates  a 
strong  presumption  in  favor  of  its  theological  truth.  It  shows 
that  Bacon's  sneer  at  barren  virgins  consecrated  to  God  is  wholly 
uncalled  for ;  here  is  a  religious  tenet,  a  purely  theological  doc- 
trine actually  giving  to  mathematicians  and  physicists  more 
knowledge  of  the  universe  than  they  could  get  from  observation 
of  nature,  without  its  aid. 


74  GEOMETRY  AND  FAITH. 

There  are  various  other  points  in  which  this  theological  hypothe- 
sis of  an  infinite  wisdom  directing  the  world  can  be  subjected  to 
tests ;  and  in  some  of  them  it  stands  the  test,  so  perfectly  as  to 
rise  rapidly  toward  a  settled  and  demonstrated  theory;  even 
upon  these  lower  grounds  of  the  understanding,  and  independent 
of  the  intuitions  of  the  higher  reason  from  which  Maupertuis  and 
the  theologians  announce  it.  Let  me  briefly  touch  upon  two  of 
them. 

In  the  early  history  of  astronomy  it  was  assumed,  that  the  law 
of  the  movement  of  the  heavenly  bodies  must  be  a  perfect  law. 
The  planet,  swinging  free  in  space,  is  subject  to  no  interruption 
from  finite  hindrances,  but  moves  under  the  influence  of  universal 
law  alone ;  therefore  its  motion  is  perfect.  This  grand  theologi- 
cal conception  is  worthy  a  place  beside  Maupertuis's  principle 
of  the  least  action.  But,  in  attempting  to  verify  this  hypothesis, 
the  ancient  astronomers  made  another  assumption  not  so  fortu- 
nate and  trustworthy ;  the  assumption  that  circular  motion  is  the 
only  perfect  motion.  They  assumed  that  the  orbit  of  a  planet  is 
circular.  But  on  putting  this  to  the  test  it  failed  to  account  for 
the  appearances  in  the  heavens.  Still  this  did  not  shake  their 
faith  in  the  perfection  of  circular  motion.  They  only  thought 
that  there  must  be  a  combination  of  circles.  Instead  of  a  single 

~ 

arm  carrying  the  planet,  they  put  an  arm  upon  the  end  of  an 
arm ;  and  conceived  it  as  rotating  twice,  while  the  first  arm  was 
rotating  once.  If  the  second  arm  be  very  short,  in  comparison 
with  the  first,  the  path  of  the  planet  carried  by  it  would  differ 
but  slightly  from  a  circle ;  while  if  the  arms  are  nearly  equal  in 
length  the  path  would  be  very  different.  But  they  soon  found 
that  no  imaginable  proportion  between  the  lengths  of  the  two 
arms  would  enable  them  to  represent  well  the  planetary  motions. 
Still  adhering  to  the  circular  movement,  they  place  a  third  arm 
at  the  end  of  the  second.  This  third  arm  was  to  revolve  three 


PHYLLOTAXIS.  75 

times,  while  the  second  revolved  twice,  and  the  first  arm  once. 
By  giving  proper  proportions  to  the  lengths  of  these  three  arras 
the  orbit  of  any  planet  could  be  described  with  tolerable  accu- 
racy. By  the  addition  of  fourth  and  fifth  arms,  to  revolve  re- 
spectively four  and  five  times,  while  the  first  revolved  once,  the 
positions  of  the  planets  for  a  single  revolution  in  their  orbits 
could  be  given  with  the  utmost  nicety. 

These  epicycles  of  Hipparchus  are  no  longer  used  in  astron- 
omy ;  but  they  have  been  used  in  modern  days,  in  other  depart- 
ments ;  in  what  might  be  called  the  statistics  of  physics,  — 
tabulated  observations.  It  has  been  shown  that  with  from  three 
to  six  arms,  rotating  each  with  a  rapidity  proportioned  to  the 
number  of  the  arm,  beginning  at  the  centre ;  the  end  of  the  outer 
arm  may  be  made  to  move  in  any  path  required,  provided  we 
may  fix  the  length  of  the  arms  as  we  choose,  and  put  them  in 
what  position  we  choose,  at  the  beginning  of  the  motion.  And 
it  is  justly  esteemed  a  grand  work  of  Hipparchus  and  of  modern 
analvsts  (including  our  own  Peirce)  that  they  should  have  invented 
and  perfected  by  two  thousand  years  of -study  so  remarkable  a 
result,  as  that  of  describing  any  outline  whatever,  by  the  simple 
device  of  rotating,  simultaneously,  three  or  more  radii  linked 
together  by  their  ends.  Yet  in  nature  a  similar  device  had  been 
used  in  a  rude  form  at  least  from  the  eozoic  ages ;  and  in  a  per- 
fected form,  even  better  than  that  of  Hipparchus  and  incompara- 
bly more  practical  and  rich  in  results,  from  the  very  advent  of 
man  upon  the  planet.  The  difference  between  the  epicycles  of 
Hipparchus  and  those  of  nature,  is  that  in  the  former,  the  ratio 
of  the  rapidities  of  rotation  is  fixed,  and  the  ratio  of  the  lengths 
of  the  rotating  arms  is  varied  at  pleasure ;  while  in  nature  it  is 
the  proportion  of  their  lengths  that  is  fixed,  and  the  proportion- 
ate velocity  of  rotation  is  varied  at  pleasure.  For  what  is  this 
right  arm  and  hand  of  man  but  a  linked  series  of  radii  in  which, 


76  GEOMETRY  AND  FAITH. 

from  the  mechanical  necessities  of  the  skeleton,  the  length  of  the 
spokes  is  fixed,  but  the  resulting  stiffness  is  much  more  than 
compensated  by  the  variable  proportions  allowed  to  the  veloci- 
ties of  rotation.  The  end,  free  motion  in  any  direction,  could  of 
course  be  attained  without  a  skeleton ;  as  for  example  the  tip  of 
the  human  tongue  can  readily  be  trained  to  move  in  any  conceiv- 
able path  within  the  buccal  chamber.  But  when  strength  and 
dignity  have  been  given  to  the  frame  by  a  skeleton,  and  mobility 
by  articulations,  the  articulations  are  exquisitely  adapted  to  the 
psychic  needs  of  the  species.  No  mechanical  necessity  can  be 
conceived  as  producing  this  adaptation ;  nor  can  I  see  the  proba- 
bility, or  even  possibility,  of  this  adaptation  having  been  pro- 
duced by  the  mutual  reaction  of  the  psyche  of  the  animal  and  its 
environment. 

In  the  case  of  man,  his  immensely  varied  mental  and  spiritual 
powers  and  capacities;  his  need  of  actions  incalculably  more 
varied  than  those  needed  by  the  brute;  whether  we  regard  his 
movements  as  aimed  to  produce  physical  or  mental  effects ;  require 
that  he  should,  have  much  greater  freedom  of  movement  under 
definite  control.  And  this  is  accomplished  for  him,  partly  through 
the  perfection  of  his  general  form,  so  "  express  and  admirable ; " 
but  more  particularly  by  the  epicycloidal  movements  of  the  hand. 
The  shoulder  is  the  centre  of  these  movements,  but  it  is  not 
rigidly  fixed,  it  has  a  proper  motion  of  its  own  by  the  sway  and 
torsion  of  the  trunk,  by  the  movement  also  of  the  lower  limbs, 
by  which  the  whole  body  has  a  capability  of  motion.  But  assum- 
ing, for  the  moment,  this  moveable  shoulder-blade  to  contain  in 
its  socket  the  centre  of  the  epicycloidal  movement  of  the  hands ; 
we  have  first  the  humerus,  swinging  freely  in  all  directions, 
within  a  cone  of  about  120° ;  which  is  accomplished  by  what  is 
called  a  ball  and  socket  joint.  The  second  link  consists  of  two 
bones  in  the  fore  arm,  hinged  to  the  first  link  by  a  hinge  joint; 


PHYLLOTAXIS.  77 

but  this  joint  receives  an  equivalent  for  ball  and  socket  freedom, 
by  the  humerus  rotating  about  90°  on  its  own  axis.  The  third 
link  is  the  metacarpus,  jointed  to  the  arm  by  a  compound  hinge, 
which,  however,  has  an  equivalent  for  ball  and  socket  freedom  by 
the  peculiar  semi-revolution  of  the  two  bones  of  the  forearm 
about  each  other.  For  ordinary  movements  these  three  links 
suffice ;  the  phalanges  being  kept  in  a  fixed  position  with  reference 
to  the  metacarpus;  just  as  three  arms  suffice  for  the  easier  curves 
in  the  Hipparchian  epicycles.  But  for  the  nicest  delicacy  there 
are  the  three  joints  of  the  phalanges,  perfecting  the  instrument, 
just  as  additional  short  radii  are  sometimes  needed  to  perfect  the 
epicycles. 

Every  one  is  familiar  with  the  fact  that  a  trained  hand  can 
sweep,  with  a  crayon,  any  outline  or  figure  whatever  upon  the 
blackboard.  What  I  am  endeavoring  to  show  is,  that  this  ability 
is  furnished  by  an  adaptation  of  the  skeleton  and  muscular  system 
to  freedom  of  motion,  and  to  freedom  under  control ;  by  a  system 
similar  to  the  Hipparchian  epicycles,  famous  in  astronomy  and 
physics.  In  fact,  every  motion  of  the  finger  ends,  so  long  as  the 
feet  remain  in  one  spot,  is  an  epicycloidal  motion ;  the  multitude 
of  links  giving  it  a  practical  infinity  of  possible  forms.  Even  the 
six  links  from  the  scapula  to  the  finger-ends  give  an  incalculable 
variety  of  possible  forms,  which  our  imagination  cannot  distin- 
guish from  an  infinite  variety.  Here  then  is  a  wisdom  and  skill  in 
the  adaptation  of  the  body  to  the  mind  of  man,  which  seems  to 
me  an  irrefragable  proof  of  the  existence  of  a  wise  designer  of  the 
animal  kingdom. 

Let  us  turn  now  to  the  vegetable  kingdom  for  the  second  in- 
stance by  which  I  would  test  the  hypothesis  of  the  world  being  the 
work  of  an  infinitely  wise  Builder.  Suppose  such  a  Builder  to 
design  a  part  of  the  vegetable  kingdom  to  grow  by  leaves  and 
buds  on  an  ascending  axis ;  the  leaves  to  be  the  lungs  and  stomach 


.'. 

78  GEOMETRY  AND  FAITH. 

of  the  plants,  and  to  require  light  for  the  fulfilment  of  their 
functions.  In  the  usual  growth  of  such  plants  it  will  be  manifest 
that  light  from  the  zenith  will  be  most  valuable.  Side  lights  will 
be  more  apt  to  be  cut  off  by  neighboring  plants  or  other  obstacles ; 
and,  even  if  not  cut  off,  may  have  and  will  be  likely  to  have  the 
highest  pitched  and  most  valuable  vibrations  absorbed  and 
destroyed  or  diverted  by  passing  horizontally  through  so  much 
more  of  the  lower  atmosphere.  The  first  necessity  of  a  leaf, 
therefore,  will  be  zenith  light.  And  it  will  be  expected  of  uner- 
ring wisdom,  we  might  almost  say  it  would  be  expected  of  the 
divine  justice,  that  each  leaf  should  have  the  fairest  possible 
chance  to  have  zenith  light.  In  other  words,  it  will  be  expected 
that  the  leaves  of  plants  should  not  be  placed  one  over  another ; 
but  they  should  be  scattered  around  the  stem  in  such  way  as  to 
give  each  the  least  shade,  and  the  most  light  possible.  Of  course 
it  will  not  be  essential  that  this  should  be  done  with  perfect  ac- 
curacy ;  practical  ends  do  not  need  theoretical  exactness ;  on  the 
contrary,  it  is  a  mark  of  a  poor  workman  to  have  him  give  a 
degree  of  finish  and  exactness  incommensurate  with  the  nature  of 
the  work ;  finishing,  for  example  a  kitchen  clock  as  one  would 
finish  a  clock  for  an  astronomical  observatory  ;  that  is  waste,  not 
economy,  of  power.  But  we  should  expect  in  plants  built  by  an 
infinitely  wise  creator  to  find  distinct  evidence  that  a  general 
plan,  for  the  accurate  distribution  of  the  leaves  around  a  vertical 
stem,  was  in  operation  ;  not  distributing  them  in  exact  conformity 
with  the  plan,  but  near  enough  for  practical  purposes ;  and  evi- 
dently showing  a  perfect  knowledge  of  the  perfect  plan. 

But  what  would  be  a  perfect  plan  ?  Like  the  old  astronomers, 
we  are  assuming  that  the  plan  of  creation  is  perfect;  let  us  not  too 
hastily  assume,  as  they  did,  that  we  know  what  a  perfect  plan 
would  be.  But,  as  near  as  we  can  see  it,  a  theoretically  perfect 
plan  ought  to  be  a  symmetrical  plan ;  to  go  on  by  a  general  law ; 


PHYLLOTAXIS.  79 

and  to  place  the  leaves  always  as  far  apart  laterally  as  possible,  as 
we  go  up  the  stem ;  so  that  they  shall  shade  each  other  as  little  as 
possible.  The  first  leaf  being  at  the  bottom,  the  second  one  going 
up  must  be  nearly  opposite,  and  thus  divide  the  circumference  of 
the  stem  nearly  in  halves.  Yet  when  the  third  leaf  comes  out  still 
higher,  the  circumference,  as  you  look  down  upon  the  plant,  must 
be  nearly  divided  into  thirds.  When  the  fourth  leaf  is  added,  the 
circumference  must  be  nearly  divided  into  quarters ;  when  the 
fifth  leaf  conies  out,  nearly  into  fifths ;  and  so  on.  The  distance 
round  the  stem,  the  angular  distance,  as  you  look  straight  down 
on  the  end  of  the  stem,  between  any  leaf  and  the  one  next  above, 
or  next  below  it,  must  evidently  be  between  y$  and  y2  a  circum- 
ference ;  or  else  between  y2  and  ^  ;  were  it  not  so,  the  first  three 
leaves  could  not  fulfil  the  requirements  of  the  problem. 

These  requirements  with  reference  to  the  first  three  leaves  can 
easily  be  put  in  a  simple  algebraical  equation.  The  second  leaf 
must  divide  the  circumference  in  halves  as  nearly  as  the  third  leaf 
makes  thirds  of  it.  That  is  to  say :  The  remainder  of  the  cir- 
cumference, after  two  angles  have  been  taken  out,  must  be  to  the 
remainder  after  only  one  was  taken,  as  that  one  was  to  the 
whole.  But  by  a  well-known  proposition  in  proportion  this  yields 
at  once  the  proportion  that  the  angle  is  to  the  remainder  of  the 
circumference,  as  that  remainder  is  to  the  whole  circumference. 
In  other  words  the  circumference  should  be  divided  in  extreme 
and  mean  ratio ;  a  curious  expression  familiar  in  geometry,  signi- 
fying the  division  of  a  thing  into  two  parts,  such  that  the  smaller 
part  is  to  the  larger  as  the  larger  is  to  the  whole.*  Division  in 
extreme  and  mean  ratio  was  invented  by  the  early  geometers, 
before  the  Christian  era,  as  a  means  of  inscribing  a  five-^sided 
figure  in  a  circle ;  but  it  was  never  suspected  that  it  occurred  in 

*  Let  the  arc  AB  be  x  ;  the  circumference,  1.  Let  BC  and  CD  each  =  AB  =  x.  That 
the  halves  AB,  and  BCA,  may  be  in  the  same  proportion  as  the  thirds  AB,  BC, 


80  GEOMETRY  AND  FAITH. 

nature  until  1849  years  after  that  era.  The  smaller  part  is  nearly 
382  thousandths  of  the  whole  ;  and,  if  any  one  wishes  to  calculate 
it  more  nicely,  he  may  extract  the  square  root  of  5  as  far  as  he 
pleases,  subtract  it  from  3,  and  divide  the  remainder  by  2.  It  is 
a  peculiar  fraction  ;  added  to  its  own  square  root  it  produces 
unity  ;  it  is  also  one-third  of  the  sum  of  1  added  to  its  square. 
But,  as  the  square  root  of  5  is  inexpressible  in  numbers,  this  pecu- 
liar fraction  is  also  inexpressible.  Its  presence  in  nature,  were  it 
there,  could  not  be  detected  by  any  microscope.  All  that  we 
could  hope  to  find  would  be  approximations  to  it.  If  a  plant  had 
such  a  law  of  growth  that  the  successive  leaves  had  a  tendency  to 
follow  each  other  at  equal  intervals,  each  .381966  of  a  circumfer- 
ence horizontally  from  its  neighbor,  we  could  not  measure  the 
angle  from  centre  to  centre  of  the  leaves,  accurately  to  any  thing 
like  the  one  millionth  ;  we  should  be  content  with  finding  it  about 
382  thousandths. 

If  the  perpendicular  distance  between  two  successive  leaves 
(called  in  botany  an  internode),  is  large,  we  can  readily  trace  a 
helical  line  around  the  stem,  passing  in  succession  through  the 
foot  of  every  leaf.  This  may  be  called  the  main  or  principal  helix. 


and  CA,  we  will  put  AC  :  CB  =  AB  :  BCA  ;  or  *-"—  =  —  x  —  , 

x          1  —  x 

By  proportion,  we  can  add  each  denominator  to  its  own 
numerator  giving  —  ^"—  =    -  ;  and  this,  by  transposing 

X  1  —  X 

the  members,  is  extreme  and  mean  ratio,  1:1  —  x  =  l  —  x:x. 
Again,  that  the  quarters  AD  and  AC  should  be  as  near 
equality  as  the  thirds,  we  put  AD  :  AC  =  AC  :  CB,  or    x 
-  ;  and  again  adding  the  denominators  {—  =  =  -    —  ;  then  adding  the  numera- 

tors to  the  denominators,  -  -  =  —  ~—  which  is  extreme  and  mean  ratio  as  before. 
1  —  x  1 

The  same  result  would  follow  at  every  step  ;  showing  that  this  length  of  step  would 
always  make  the  nearest  approach  possible  to  equal  division  iiirto  as  many  parts  as  there 
had  been  points  placed,  whatever  the  number  of  steps. 


PIIYLLOTAXIS.  81 

And  it  is  evident  that  we  may  take  the  very  same  set  of  leaves 
and  pass  a  helix  in  an  opposite  direction  through  every  leaf ;  only 
in  that  case,  the  angular  distance  between  two  leaves  will  be  618 
instead  of  382  thousandths  of  the  way  round. 

Let  us  now  inquire,  what  further  visible  and  easily  observable 
evidence  we  should  have,  if  a  plant  was  actually  constituted  with 
this  law  of  extreme  and  mean  ratio,  as  the  ideal  plan  of  its  distri- 
bution of  leaves ;  as  seems  to  be  demanded  by  perfect  wisdom  and 
justice,  and  by  symmetry.  In  case  of  the  internodes  being  com- 
paratively short,  this  principal  helix  will  wind  round  the  stem 
with  its  threads  so  close,  and  the  leaves  so  crowded  that  it  will  be 
difficult  for  the  eye  to  follow  the  helix,  or  discover  order  in  the 
arrangement  of  the  leaves.  Let  us  imagine  the  leaves  on  a  piece 
of  stem  arranged  in  this  ideal  order,  and  numbered  from  zero, 
upward  in  the  order  of  the  stem.  Split  down  the  stem  on  the 
opposite  side  to  the  zero  leaf,  take  off  the  bark,  and  spread  it  flat. 
If  the  internodes  are  short,  the  numbers  over  the  zero  will  be 
arranged  as  those  in  figure  A  are  arranged,  when  you  turn  the  fig- 
ure cornerwise,  so  as  to  have  0  at  the  bottom,  and  55  almost  over 
it,  a  shade  to  the  right  of  the  vertical.  If  the  internodes  are 
longer,  the  numbers  are  better  represented  by  B. 

FIGURE  A. 

15  23  31  39  47  55 
10  18  26  34  42  50 
5  13  21  29  37  45 
0  8  16  24  32 
.  3  11  19 
6  14 


82  GEOMETRY  AND  FAITH. 

FIGURE  B. 

15    23    31    39    47    55 
10     18     26    34    42    50 
5     13     21     29    37 
0     8     16    24    32 
3     11     19    27 
6     14    22 

In  either  figure  the  principal  helix,  joining  consecutive  num- 
bers, is  transformed  into  a  series  of  parallel  straight  lines.  But 
we  find  also  that  straight  lines  from  any  selected  leaf  in  any  direc- 
tion, join  equidistant  numbers ;  and  every  straight  line,  when  the 
bark  is  rolled  back  into  its  cylindrical  form,  will  become  a  helix. 
Thus  the  lines  in  figure  A,  joining  3  to  5,  6  to  8,  and  10,  14  to  16 
and  18,  etc.,  would  become  two  secondary  helices,  steeper  than 
the  principal  one. 

Again,  it  will  be  observed  that  the  numbers  0,  3,  6,  etc.,  5,  8, 11, 
14,  etc.,  are  in  parallel  straight  lines ;  and  would  make  still  steeper 
tertiary  helices  on  the  stem. 

Turning  once  more  in  the  opposite  direction,  we  find  yet  steeper 
parallel  lines  joining  every  fifth  number;  such  as  0,5,  10,  etc.;  3, 
8,  13,  etc. 

Still  closer  to  the  perpendicular,  and  in  the  direction  of  the 
original  helix,  we  trace  the  lines  0,  8,  16  and  5,  13,  21,  joining 
every  eighth  leaf. 

With  a  still  greater  compression  of  the  length  of  the  stem,  and 
a  diminution  of  the  size  of  the  numbers,  lines  would  become  con- 
spicuous, joining  the  leaves  0,  13,  26,  etc.;  or  the  leaves  0,  21,  42; 
with  very  great  compression  and  very  small  numbers,  even  the 


PHYLLOTAXIS.  83 

lines  0,  34,  68,  and  0,  55,  110,  almost  perpendicular,  would  become 
visible  and  conspicuous. 

Observe  how  curiously  these  numbers  are  generated  from  each 
other.  Starting  from  zero  on  B,  to  go  straight  to  3  you  pass  be- 
tween 1  and  2 ;  but  1  +2  =  3.  Passing  from  0  to  5  we  go  close 
in  between  2  and  3,  and  2  +  3  =  5.  Again  the  passage  from  0  to 
8  is  between  5  and  3,  and  5  +  3  =  8.  The  gateway  from  0  to  13 
is  between  8  and  5  ;  but  8  +  5  =  13. 

Observe,  also,  how  readily  these  auxiliary  helices  will  enable  us 
to  number  the  leaves.  Suppose  this  piece  of  bark,  with  its  foot- 
prints of  the  leaves  on  it,  is  before  us,  but  the  leaves  not  numbered. 
Assume  any  one  as  zero.  Now  here  are  three  parallel  helices 
taking  in  every  leaf.  Then  upon  the  one  passing  through  0  you 
can  certainly  write  the  numbers  0,  3,  6,  etc.  Here  are  five  other 
helices  crossing  the  first  and  including  every  leaf.  Then  upon  the 
one  passing  through  0  you  can  write  0,  5,  10,  15,  etc.  You  can 
now  pass  to  any  leaf  whatever  by  adding  or  subtracting  5  or  3, 
as  you  move  xip  or  down  on  either  one  of  either  set  of  helices. 

Were  the  stem  crowded  with  a  greater  multitude  of  leaves,  so 
that  the  more  perpendicular  and  more  numerous  helices  were  most 
prominent,  the  same  thing  could  be  done.  For  example,  were 
there  21  helices  in  one  direction  and  34  in  the  other,  and  you 
wished  to  find  the  8th  leaf  from  one  which  you  had  chosen  as 
zero,  you  have  simply  to  ascend  two  steps,  on  one  of  21  leaf  lines, 
and  descend  one  step  on  the  34  leaf  line ;  since  42  —  34  =  8. 

It  will  at  once  be  seen  that  if  the  stem  of  a  plant  be  tough,  and 
of  even  texture,  so  that  it  will  bear  twisting  evenly ;  then,  if  the 
leaves  were  arranged  in  this  fashion,  a  twist  in  the  stem  would 
straighten  up  every  alternate  set  of  helices,  and  flatten  down  the 
others.  A  very  gentle  twist  might,  for  example,  bring  the  34th 
leaf  over  the  zero.  Every  leaf,  the  twist  being  uniform,  would 
then  be  exactly  over  the  34th  below  it.  The  34  leaf  helices  would 


84  GEOMETRY  AND  FAITH. 

be  vertical  lines,  and  the  leaves  would  be  in  34  perpendicular 
rows.  A  little  harder  twist  would  bring  the  13th  leaf  over  the 
zero ;  and  the  leaves  would  then  stand  in  13  vertical  rows.  Twist 
still  harder;  the  5th  leaf  comes  over  the  zero;  the  five  helices  be- 
come five  vertical  rows.  If  your  stem  is  tough  enough,  twist  vio- 
lently enough  to  bring  2  over  the  0,  and  your  leaves  are  in  two 
rows,  alternately  opposite. 

Let  us  now  imagine  our  gentle  twist  at  the  beginning  to  be  in 
the  opposite  direction,  and  bring  the  21st  leaf  over  the  zero ;  this 
would  give  us  21  perpendicular  rows ;  a  little  more  twist  in  that 
direction  would  develop  8  rows ;  a  hard  twist  reduce  them  to 
three. 

This  supposed  twisting  of  the  stem  would  increase  or  diminish 
the  theoretically  perfect  angle,  .381966  of  the  circumference, 
until,  in  the  case  of  the  two  rows,  the  angle  was  .5  ;  or,  in  the 
case  of  the  three  rows,  it  was  .333333.  Between  these  ex- 
tremes, y?  and  YJ,  of  the  circumference,  lie  all  those  arithmetical 
approximations  to  the  perfect  angle,  which  we  have  developed  by 
twisting ;  1  : 2,  1 :  3,  2  : 5,  3  :  8,  5  : 13,  8  :  21,  13 : 34,  21  :  55,  34  :  89, 
55  : 144,  89 : 233,  144 : 377.  These  approximate  angles  are  formed 
each  from  the  two  preceding,  by  simply  adding  the  two  numera- 
tors, and  adding  the  two  denominators. 

One  half  is  larger  than  .381966,  and  y$  is  smaller;  and  through- 
out the  series  they  are  alternately  larger  and  smaller  than  the 
perfect  angle.  No  other  vulgar  fraction  stands  in  the  line  of 
direct  advance.  Take  for  example  7  : 19,  it  is  not  as  near  as  3  :  8, 
nor  is  7  : 20,  nor  5  : 14,  nor  any  fraction  with  a  denominator  under 
21,  except  5:13.  This  series  settles  toward  the  extreme  and 
mean  ratio  like  a  vibrating  needle  settling  to  the  magnetic  merid- 
ian ;  the  last  one  given  above  differs  from  the  true  angle  by  only 
.001  of  1°. 

Suppose  we  should  now  take  the  case  of  an  actual  stem  crowded 


PHYLLOTAXIS.  85 

with  leaves,  and  should  find  that  an  actual  twist  to  the  left  or  to 
the  right  would  bring  the  leaves  into  3,  5,  or  8  rows,  what  would 
it  prove  ?  From  the  diagrams  A  and  B  it  would  prove  that  the 
leaves  were  actually  arranged  at  equal  angles  around  the  stem,  and 
that  the  angle  was  between  1 : 3  and  2 :  5  of  a  circumference.  Sup- 
pose, however,  that  a  slighter  twist  would  bring  the  leaves  into  13 
or  21  rows ;  or  suppose  that  the  34  and  55  leaf  helices  were  already 
conspicuous,  without  any  twist ;  and  suppose  that  by  counting  up 
by  34s  and  55s  we  should  find  the  377th  leaf  was,  more  exactly 
than  any  other,  vertical  over  the  zero  leaf.  This  would  prove  the 
leaves  to  be  actually  arranged  not  only  evenly  around  the  stem, 
but  at  an  angle  almost  precisely  that  of  the  theory. 

These  suppositions  are  actually  verified.  Taking  up,  for  ex- 
ample, a  stalk  of  broad-leaved  plantain,  crowded  with  flower  buds, 
or  with  seed  pods ;  you  can,  by  twisting  in  one  direction,  bring 
out  8  or  3  rows ;  by  twisting  in  the  other,  5  rows,  and  with  a  hard 
twist,  2.  Taking  up  any  pine  or  spruce  cone,  and  numbering  a 
few  scales  by  first  counting  the  conspicuous  parallel  helices,  in 
each  of  the  two  conspicuous  intersecting  sets,  you  will  find  the 
13th,  21st,  34th,  or  55th  leaf  come  most  directly  over  the  leaf 
taken  as  your  zero. 

When  the  cylindrical  stem  is  contracted  into  a  strobile,  cyme  or 
rosette,  or  into  the  nearly  flat  head  of  a  composite  flower,  the 
helices  are  transformed  into  other  curves.  Were  the  stem  trans- 
parent, and  the  helices  drawn  in  perspective  upon  a  plane  at  right 
angles  to  it,  the  eye  being  the  axis  of  the  stem,  the  helices  would 
become  hyperbolic  spirals.  But  the  spirals  of  sunflower  seeds 
make  no  approach  to  that  form.  Were  a  short  portion  of  the 
stem,  the  height  being  equal  to  the  radius,  folded  in  toward  the 
centre,  the  distance  of  the  helix  from  the  circumference  being  un- 
changed, the  helix  would  become  a  spiral  of  Conon,  or  Archimedes. 
But  the  sunflower  does  not  conform  to  that  law. 


86  GEOMETRY  AND  FAITH. 

The  helices  each  make  a  constant  angle  with  the  meridian  of 
the  cylindrical  stem ;  and  as  far  as  I  have  yet  observed,  this  is 
true  of  the  transformed  curves  ;  so  that  on  a  globular  strobile  the 
helix  becomes  a  rhumb  line  ;  and  on  the  sunflower  head,  a  spira 
mirabilis  of  Bernouilli.  In  the  rosettes  of  young  ^Enotheras,  Cap- 
sellas,  etc.,  the  spiral  is  developed  by  a  different  process  ;  and  may 
possibly  be  a  different  spiral,  but  the  sunflower  certainly  approxi- 
mates (according  to  my  measurement)  very  closely  to  a  logarith- 
mic spiral ;  the  spira  mirabilis. 

In  the  heads  of  composite  flowers  these  spirals  are  beautifully 
conspicuous,  and  afford  an  easy  method  of  determining  the  de- 
gree of  approximation.  The  vertical  lines  have  here  become  radii. 
There  will  always  be  two  spirals  (one  running  in  each  direction), 
plainly  to  be  seen,  crossing  at  each  seed.  The  value  of  each  of 
these  can  be  determined  by  counting  round  the  stem  and  seeing 
how  many  similar  spirals  there  are  in  each  direction.  Thus,  if 
there  are  8  spirals,  as  it  were,  parallel  to  each  other,  running  round 
to  the  right,  there  they  must  be  the  8  leaf  helices ;  and  the  5 
leaf  helices  will  be  found  running  to  the  left,  5  in  number.  As- 
suming now  any  seed  as  a  zero  leaf,  the  adjacent  seeds  running 
up  the  spiral  to  the  right  will  be  the  8th,  16th,  24th,  etc. ;  and 
running  to  the  left  the  5th,  10th,  15th,  etc.  By  running  to  the 
left  and  right  alternately  you  can  thus  determine  the  number  of 
any  seed  as  you  please.  By  determining  as  nearly  as  possible  the 
centre  of  the  head,  you  can  draw  a  radius,  from  your  assumed 
zero  and  inward.  If  that  radius,  after  leaving  the  zero,  strikes 
first  on  the  centre  of  number  34,  the  angle  is  13  : 34 ;  but  if  it 
steers  between  34  and  55,  grazes  89,  and  strikes  a  seed  centrally 
first  at  144,  then  the  angle  must  be  55 : 144. 

As  I  was  writing  this,  I  picked  up  three  heads  of  dandelion  in 
seed,  blew  off  the  seeds,  and  counted  the  pits  in  the  receptacle. 
One  of  them  was  on  the  34 :  89  arrangement ;  the  other  two  each 


PIIYLLOTAXIS.  87 

on  the  55  : 144  arrangement.  The  largest  of  these  was  half  an  inch 
across ;  and  the  error  in  the  position  of  any  two  consecutive  seeds 
in  the  outer  row  was,  therefore,  less  than  one  30,000th  of  an  inch. 
One  of  the  heads  gave  an  angle  a  trifle  too  large  ;  the  other  two, 
an  angle  a  little  too  small ;  the  average  was  almost  exact. 

Subsequently,  I  counted  the  first  ox-eye  daisy,  and  the  first  sun- 
flower which  I  saw.  They  were  equally  exact  as  the  dandelion, 
—  the  sunflower  even  nearer.  It  was  upon  a  144:377  approxi- 
mation ;  that  is,  the  angle,  instead  of  being  137°.508,  was  137°.507, 
only  about  a  thousandth  of  a  degree  too  small.  The  head  was 
about  20  centimetres  or  8  inches  in  diameter ;  so  that  two  con- 
secutive seeds  near  the  circumference  would  have  an  arc  of  9^ 
inches  or  24  centimetres  between ;  and  this  arc  was  actually  about 
one  500th  of  a  millimetre,  less  than  the  ten  thousandth  of  an  inch, 
too  small.  It  would  surely  be  unreasonable  to  ask  for  any  closer 
conformity  of  observation  to  theory. 

Xor  is  it  easy  to  imagine  any  cause  which  necessitates  the  ar- 
rangement. It  has  been  shown  that  if  a  cell  generate  cells  on  a 
horizontal  plane  at  equal  intervals  of  time  ;  and  each  cell  begin  to 
generate  in  the  same  manner,  as  soon  as  it  is  two  intervals  old : 
and  the  generation  be  always  on  a  plane,  and  at  the  right  hand 
side ;  then  the  cells  will  be  arranged  like  the  seed  pits  in  the  dan- 
delion receptacle.  But  this  goes  very  little  way, — nay,  I  do  not 
see  that  it  even  starts  on  the  road,  — toward  showing  us  the  gen- 
esis of  the  ascending  helix. 

Again  it  has  been  suggested  that  as  leaves  grow  by  light  and 
air,  they  will  naturally  grow  where  they  have  the  best  chance  at 
getting  them ;  and  this  in  the  course  of  generations  would  lead 
them  to  come  out  exactly  at  the  right  spot.  Unfortunately  for 
that  explanation,  the  leaves  of  plants  which  need  the  light  and 
air,  and  for  whose  benefit  we  have  invented  the  law,  do  not  con- 
form to  it  in  such  wise  as  to  make  it  a  physical  benefit.  They 


88  GEOMETRY  AND  FAITH. 

grow  almost  universally  on  the  lowest  approximations,  2,  3,  and  5 
ranked. 

Practically,  the  physical  benefit  is  not  felt.  And  in  the  part 
where  the  highest  approximations  are  reached,  in  the  heads,  cones, 
cymes,  etc.,  the  physical  benefit  is  lost  through  the  crowding.  We 
have  evidently  over-estimated,  at  the  beginning  of  our  speculation, 
the  merely  physical  necessity  of  the  arrangement. 

Of  course  we  can  set  no  bounds  to  the  discovery  of  physical 
causes  for  physical  effects ;  and  it  is  therefore  possible  that  the 
botanist  may,  at  some  day,  discover  the  physical  agencies  by 
which  this  physical  arrangement  of  leaves  is  effected.  But  when 
he  has  done  so,  he  will  not  have  in  the  least  shaken  the  theological 
inferences.  The  preponderance  of  the  ruder  approximations, 
1:2,  1 :3,  2:5,  and  3:8,  (1:3  and  3:8  giving  too  smally  and  1 :2 
and  2  : 5  too  large,  an  angle)  shows  that  the  perfect  phyllotactic 
law  is  not  of  practical  importance  in  the  growth  of  plants ;  they 
live  and  flourish  on  the  rudest  approaches  to  it.  But  the  tracing 
of  these  approximations  up,  in  such  very  numerous  instances,  to 
the  highest  degree  of  accuracy,  such  as  55  : 44  and  34 : 89,  one 
above,  the  other  below  the  perfect,  shows  that  the  law  of  extreme 
and  mean  ratio  is  actually  incorporated  into  the  vegetable  king- 
dom. The  builder  of  the  plant  knew  that  law  untold  ages  before 
the  geometer  invented  it,  to  inscribe  a  pentagon.  These  succes- 
sive approximations  point  out  more  clearly  and  strikingly  than 
absolute  conformity  to  it  could  have  done. 

And  as  its  efficient  cause  thus  lay  in  the  divine  wisdom  and 
divine  power,  so  its  final  cause  lies  also  in  the  spiritual  realm. 
We  have  come  upon  it  by  an  assumption  that  the  leaves  are 
treated  justly ;  that  each  is  given  the  best  possible  chance  at  light 
and  air.  But  while  we  have  learned  from  our  examination  of  it 
that  the  divine  Architect  knew  this  need  of  the  leaf,  and  in  pro- 
viding for  it  took  this  absolutely  perfect  law ;  we  learn  also  that 


PHYLLOTAXIS.  89 

he  knew  that  perfect  conformity  was  not  physically  needed  ;  and 
he  therefore  allowed  these  continued  and  great  variations  by 
which  the  law  is  suggested  rather  than  thrust  upon  us ;  he  made 
the  symmetry  of  the  plant  potential,  rather  than  actual,  and  this 
suggestion,  rather  than  actualization,  of  the  perfect,  makes  the 
plant  a  more  valuable  teacher  and  companion  of  man.  The  sug- 
gestion of  infinite  perfection,  that  is  beauty. 

The  Lethe  of  nature 

Cannot  trance  him  a<rain, 
Whose  soul  sees  the  perfect, 

Which  his  eyes  seek  in  vain. 

The  outward  eye  cannot  directly  see  the  division  of  the  circum- 
ference in  extreme  and  mean  ratio ;  half  the  leaves  are  hidden, 
and  even  if  we  see  two  consecutive  leaves,  we  cannot  tell  the 
precise  angle.  But  the  secondary  helices  of  approximation  are 
constantly  visible,  and  give  a  great  geometric  fascination  to  the 
fructification,  sometimes  even  to  the  foliage ;  while  in  the  larger 
growth  of  the  plant  the  law  secures  general  symmetry ;  the  varia- 
tion and  concealment  through  various  causes,  prevent  monotony 
and  give  an  endless  charm  of  variety. 


XV. 

NUMBER  AND  PROPORTION. 

IT  is  only  at  a  comparatively  late  period,  in  the  development 
of  the  human  mind,  that  number  comes  into  view,  as  a  distinct 
object  of  thought.  The  idea  of  number  is  evolved  from  things 
of  imperfect  unity ;  as  an  abstraction  from  things  concrete,  tan- 
gible and  audible.  The  two  hands,  the  ten  fingers,  the  mother, 
the  nurse,  the  window-panes,  suggest  the  idea ;  and  it  is  slowly 
brought  into  the  field  of  distinct  intellection,  by  more  or  less 
laborious  effort.  A  child  usually  attains  the  age  of  three  or  four 
years,  before  it  gives  evidence  of  attaching  clear  ideas  to  the 
names  of  numbers.  Not  until  adult  life  do  men  usually  perceive 
that  persons  are  examples  of  the  most  complete  and  absolute 
unity. 

But  this  slowness  with  which  the  idea  of  number  rises  to  the 
surface  of  consciousness,  only  shows  how  very  deeply  it  is  im- 
bedded in  the  soul.  At  the  beginning  of  conscious  life  our  atten- 
tion is  fixed  upon  the  individual  objects  presented  in  sensation. 
The  child  at  that  period, 

Nescio  quid  meditans  nugarum :  totus  in  illis : 

thinks  only  of  the  direct  lessons  of  the  outward  world.  The 
abstraction  of  number,  and  the  invisible  realities  of  space,  of 
time,  and  of  the  spiritual  world  escape  his  attention,  until  he 
arrives  at  a  mature  condition.  But  the  mature  mind  perceives 
that  this  historical  order  of  the  development  of  ideas  is  almost 
invariably  precisely  the  reverse  of  the  logical  order  of  their  de- 

90 


NUMBER  AND  PROPORTION.  91 

penclence.  For  example,  space  and  time  seem  at  first  to  be 
abstractions  from  the  observed  facts  of  matter  and  motion  ;  and 
it  is  hastily  assumed  that  the  experience  of  the  outward  world 
conies,  at  first,  independently  of  any  perception  of  space  and 
time;  and  that  these  ideas  are  derived  from  that  experience. 
This  may  be  true  in  the  chronological,  or  historical,  succession  of 
our  distinct  analytical  attention  to  the  ideas ;  the  actual  sequence 
of  distinct  conscious  attention  is,  that  we  first  perceive  motion,  or 
rather  matter  in  motion  ;  and  this  leads  us  to  the  consideration  of 
space  and  time. 

Subsequent  thought  will,  however,  always  show  that,  in  the 
very  first  perception  of  matter  in  motion,  we  quietly  take  for 
granted  the  existence  of  space,  occupied  by  matter ;  and  of  time 
taken  up  in  the  motion.  The  ideas  of  matter,  motion,  space  and 
time,  actually  enter  the  field  of  consciousness  at  the  same  instant ; 
that  is,  the  historical  order  only  relates  to  the  sequence  of  the  acts 
of  attention,  by  which  we  separate  the  ideas  from  each  other. 
In  strict  logical  connection,  the  conception  of  space  and  time  must 
precede  the  conceptions  of  matter  and  motion  ;  since  space  and 
time  are  the  conditions  on  which  motion  is  alone  possible. 

In  like  manner  it  will  be  found  that  although  number  is  a  late 
object  of  conscious  attention  ;  and  can  be  developed  as  an  object  of 
distinct  consecutive  thought  only  in  a  mind  of  some  maturity ;  it 
nevertheless  stands  logically  antecedent  to  every  act  of  intellec- 
tion. The  conscious  subject  is  conscious  of  an  object ;  and  these 
with  the  act  of  consciousness  form  a  tri-unity  at  the  very  begin- 
ning of  any  conscious  life.  If  we  dare  venture  so  high  a  flight  of 
thought,  we  may  even  say  that  as  the  creative  Mind  must  be  pos- 
ited as  a  logical  antecedent  to  creation ;  so  even  in  that  infinite 
Mind,  considered  even  as  its  own  object,  that  same  tri-unity  ex- 
isted antecedent  to  any  creation.  Of  course  we  speak  of  logical, 
and  not  of  chronological  antecedence.  Whether  the  latter  ever 


92  GEOMETRY  AND  FAITH. 

really  existed ;  whether  there  was  creation,  in  the  widest  sense  of  the 
word,  is  a  matter  beyond  our  most  daring  flight.  But  in  the  first 
act  of  intellection,  the  conception  of  subject  and  object  implies 
the  conception  of  number.  As  with  the  idea  of  self  so  with  the 
idea  of  space  and  time ;  they  also  necessarily  imply  number  the 
moment  that  they  are  made  objects  of  thought.  Even  infinite 
space,  although  absolutely  homogeneous  and  without  distinction 
of  parts,  except  as  such  parts  are  created  by  thought,  has  its  two 
elements  of  distance  and  direction  ;  and  the  element  of  direction 
has  a  manifoldnessj  which  by  no  artifice  of  ingenuity  can  be  re- 
duced to  less  than  three  dimensions.  Some  modern  geometers, 
assuming  that  the  conception  of  space  is  derived  from,  as  well  as 
suggested  by  experience,  have  speculated  upon  the  possibility 
that,  in  those  parts  of  the  universe  which  are  beyond  the  range 
of  our  experience,  space  may  have  other  properties ;  more  dimen- 
sions than  three,  or  a  curvature  by  which  two  straight  lines  might 
include  a  surface ;  and  so  on.  Pursuing  this  speculation  they 
have  investigated  certain  algebraical  sentences,  expressing  these 
impossible  conceptions ;  and  have  found  the  language  capable  of 
self-consistent  interpretation.  They  have  urged  this  possibility  of 
self-consistent  interpretation  as  an  evidence  that  the  conceptions 
themselves  may  be  realized  in  the  infinite  distance.  In  spite  of 
this  ingenuity  Reason  sturdily  maintains  that  the  properties  of 
space,  as  we  see  it  here,  are  invariable  throughout  the  whole  ex- 
tent of  absolutely  boundless  distances ;  and  although  we  may 
technically  express,  in  algebraical  language,  the  conception  of  a 
circle  having  unequal  diameters,  and  deduce  logically  self-consist- 
ent results  from  the  conception,  yet  we  can  neither  make  any 
picture  of  such  a  circle  in  our  imagination,  nor  believe  that  such 
a  circle  can  exist  in  any  regions  beyond  the  telescope. 

Number,  inhering  in  the  primal  act  of  consciousness,  follows 
every  step  of  thought.     All  intellection,  all  thinking  is  the  per- 


NUMBER  AND  PROPORTION.  93 

ception  or  creation  of  differences  and  distinctions,  unities  and 
resemblances.  The  definition  of  chemistry,  that  it  is  the  identifi- 
cation of  the  one  in  the  many,  and  the  detection  of  the  many  in 
the  one,  may  be  considered  also  as  a  definition  of  all  science  and 
of  all  thought.  Number  is  more  prominent  in  Chemistry,  just  as 
Space  is  more  prominent  in  Mechanics,  and  Time  in  Biology ;  but 
Number,  Space  and  Time  are  all  three  involved  in  every  finite  act 
of  intellection.  All  language  bears  witness  to  this  presence  of  the 
three  ideas  in  every  thought ;  take  any  word  of  any  language 
and  analyze  it  carefully,  trace  back  its  history  and  you  find  in  it 
some  more  or  less  apparent  reference  to  number  and  motion,  taken 
perhaps  as  typical  of  spiritual  things. 

But  number,  although  thus  involved  in  every  act  of  conscious- 
ness, even  the  primal,  is  not  the  highest  genus ;  it  is  a  species  of 
relation.  The  highest  unity  is  the  person  ;  and  the  highest  Per- 
son, although  we  speak  of  Him  as  Absolute  and  Unconditioned, 
stands  logically  related  to  His  own  attributes  and  to  His  own  cre- 
ation. Every  act  of  finite  intellection  involves  not  only  the  per- 
ception of  number,  but  of  other  relations  also.  There  is  more  in 
the  consciousness  of  subject,  object,  and  relation  between  them, 
than  the  mere  perception  of  tri-unity  in  the  act.  There  is  the 
perception  of  more  than  the  numbers  three,  two,  and  one.  Of 
course  in  the  more  complex  acts  of  intellect,  it  is  still  more  em- 
phatically true,  that  the  perception  of  numerical  relation  does  not 
constitute  the  whole  contents  of  consciousness.  Number  is,  how- 
ever, the  relation  by  which  the  relation  of  quantity  becomes 
amenable  to  thought  and  calculation. 

At  first,  number  is  generated  by  the  distinction  of  things  dis- 
crete, and  perhaps  different  in  kind.  It  immediately  becomes  in 
itself  an  object  of  thought ;  and  its  distinction  of  many  or  few 
becomes  the  most  firmly  grasped  and  clearly  comprehended  of  all 
relations  of  quantity ;  which  is  at  once  applied  to  the  measure- 


94  GEOMETRY  AND  FAITH. 

ment  of  all  things  that  may  naturally  be  counted.  But  there  are 
many  kinds  of  quantity  which  are,  absolutely  or  relatively,  contin- 
uous ;  and  the  measurement  of  the  greater  or  less  in  these  kinds 
is  accomplished  by  the  analogy  of  the  greater  or  less  quantity  to 
large  or  small  numbers.  So  indispensable  to  all  clear  intellection 
is  this  relation  of  numbers  to  each  other,  that  the  Greeks  called  it 
AO-/OC,  that  is  word  or  wisdom ;  and  the  Latins  called  it  ratio,  or 
reason  ;  and  this  is  its  technical  name  among  mathematicians  to 
the  present  hour.  The  ratio  of  two  numbers  is  their  relation  of 
magnitude,  not  as  estimated  by  the  excess  of  one  over  the  other  ; 
but  estimated  by  how  many  times  one  is  larger  than  the  other. 
Thus  the  excess  of  6  over  2  is  the  same  as  that  of  8  over  4 ;  but 
the  ratio  of  6  to  2,  is  3  to  1 ;  and  that  of  8  to  4  is  but  2  to  1. 
When  we  seek  to  find  the  ratio  of  one  continuous  quantity  to 
another  of  the  same  kind,  we  simply  seek  to  find  two  numbers 
having  the  same  ratio  as  the  two  quantities.  One  quantity  is  con- 
sidered as  the  unit,  to  which  to  refer  the  other.  Usually  as  a  pre- 
liminary step,  both  are  first  referred  to  some  artificial  unit ;  such 
as  an  inch,  a  meter,  a  quart,  a  liter,  a  degree  of  the  thermometer, 
a  degree  of  angle,  an  hour,  a  dollar,  etc.,  etc. 

Yet  many  of  the  most  interesting  ratios  are  found  not  to  be 
equal  to  the  ratio  of  any  two  numbers  whatever.  For  example, 
the  diagonal  and  side  of  a  square,  although  nearly  in  the  propor- 
tion of  17  to  12  are  not  exactly  in  the  ratio  of  any  two  numbers 
whatever.  And  in  general  we  may  say  that  the  mean  proportional 
between  1  and  another  number,  will  very  seldom  be  in  a  ratio  ex- 
pressible in  numbers.  Two  is  a  mean  proportional  between  1  and  4 ; 
because  1  is  to  2,  as  2  is  to  4  ;  3  is  a  mean  proportional  between  1 
and  9 ;  and  so  on.  This  mean  proportional,  or  geometric  mean, 
may  be  illustrated  by  letting  fall  a  perpendicular  from  any  point 
in  a  semi-circumference  upon  the  diameter ;  the  length  of  the 
perpendicular  is  then  a  mean  proportional  between  the  two  parts 
into  which  it  divides  the  diameter. 


NUMBER  AND  PROPORTION.  95 

In  the  attempt  to  measure  quantities,  it  is  frequently  necessary 
to  assume  a  starting  point,  and  measure  in  both  directions ;  — 
one  is  then  able  in  subtracting  to  obtain  quantities  less  than  noth- 
ing ;  like  south  latitude,  east  longitude,  temperature  below  zero, 
deficit  in  a  treasury ;  such  quantities  are  called  negative  quanti- 
ties ;  they  lead  us  at  once  to  the  useful  fiction  of  negative  num- 
bers. But  this  fiction  requires  the  additional  fiction  of  negative 
ratios.  For  example,  the  ratio  of  20  above  zero  to  10  below  zero, 
is  not  simply  that  of  2  to  1,  which  would  only  give  10  above.  It 
must  be  expressed  by  saying  it  is  negative  2  ;  —  meaning  that  it 
is  twice  as  far  from  the  zero  mark,  but  in  the  other  direction. 

But  out  of  this  comes  a  very  remarkable  case  of  impossibility  ; 
namely  the  impossibility  of  finding  the  geometric  mean  between 
a  positive  and  a  negative  number.  The  geometric  mean  between 
1  and  2  cannot  be  expressed  in  numbers ;  but  the  fraction  -}-£  is 
nearly  it ;  and  no  error  can  be  named  so  small  that  a  fraction  can- 
not be  named  differing  from  the  geometric  mean  less  than  that 
error.  But  the  geometric  mean  between  1  above  and  1  below 
zero  cannot  be  expressed  by  any  degree  of  the  thermometer  or  by 
any  conceivable  numbers.  It  must  be  1,  but  it  can  be  neither  pos- 
itive 1  nor  negative  1 ;  since  1  is  not  to  positive  1,  as  positive  1  is 
to  negative  1 ;  neither  is  1  to  negative  1,  as  negative  1  is  to  nega- 
tive 1.  Certainly  zero  is  not  the  geometric  mean  ;  for  zero  is  not 
as  many  times  negative  1,  as  1  is  times  zero.  What  then  is  this 
temperature  neither  above,  below,  nor  at  zero?  this  latitude 
neither  north  nor  south  of  the  equator  ?  It  is  called  in  mathemat- 
ics (lucus  a  non  lucendo}  the  imaginary ;  because  it  is  unimagina- 
ble. The  mathematicians  have  various  symbols  and  various 
names  for  it,  and  use  it  freely  in  their  calculations.  They  have 
endeavored,  with  great  success,  to  reduce  all  quantitative  and 
geometric  impossibilities  to  this  one ;  that  is  to  say,  they  will  give 
you  a  correct  answer  to  any  absurd  question  you  may  devise, 


96  GEOMETRY  AND  FAITH. 

provided  you  allow  them  to  introduce  into  their  answers  a  sym- 
bol standing  for  the  unimaginable  mean  proportional  between  pos- 
itive and  negative  unity.  In  geometrical  questions  this  symbol 
may  often  be  interpreted  as  signifying  the  rotation  of  a  line  about 
some  point,  but  no  general  interpretation  has  been  discovered.  It 
is,  however,  a  singular  tribute  to  the  ingenuity  of  the  mathema- 
tician, that  he  has  reduced  all  absurdities,  all  impossibilities,  to 
this  one, — the  finding  of  an  operation  upon  a  quantity  which 
shall  neither  increase  it,  diminish  it,  nor  leave  it  unchanged. 
Give  him  a  symbol,  say  i,  for  that  operation,  and  he  is  as  compe- 
tent to  deal  with  the  impossible,  as  with  the  possible.  But  the 
advantage  and  power,  gained  by  the  use  of  £,  extends  also  into 
the  realm  of  the  possible  and  of  the  actual ;  and  enables  the  math- 
ematician to  assist  in  delicate  and  complicated  researches  of  mod- 
ern physical  science,  otherwise  beyond  the  reach  of  man. 

If  we  multiply  1  by  1.00000001,  one  hundred  million  times,  the 
product  is  2.7182818.  It  is  the  second  of  three  famous  quantities, 
pertaining  to  number  alone,  yet  incapable  of  exact  expression  in 
number ;  since  exact  expression  would  require  the  decimal  part 
of  the  constant  multiplier  to  be  infinitely  small,  and  the  number 
of  multiplications  to  be  infinitely  great. 

This  peculiar  number  is  known  in  mathematics  by  the  name  of  the 
base  ;  (THE  base) ;  and  may  be  symbolized  by  b.  It  may  be  defined 
as  the  product  of  100,000,000  factors,  each  equal  to  1 .00,000,001 ; 
but  is  susceptible  of  various  other  definitions.  Like  *,  it  furnishes 
the  key  to  many  physical  problems,  otherwise  insoluble  ;  and  it  is 
very  difficult  to  express,  in  terms  not  technical,  any  reason  why  it 
should  exert  this  beneficent  power. 

The  third  of  these  famous  quantities  is  the  ratio  of  the  circum- 
ference to  its  diameter ;  which  is  nearly  22  to  7,  still  more  nearly 
355  to  113  ;  but  which  cannot  be  expressed  exactly  either  in  num- 
bers, or  in  mean  proportionals  between  numbers  ;  it  is  a  peculiar, 


NUMBER  AND  PROPORTION.  97 

unique  ratio ;  it  may  be  symbolized,  if  you  please,  by  c ;  and  its 
immense  value  in  calculation  arises  partly  from  the  fact  that  the 
difficulty  of  measuring  not  only  circles,  but  every  conceivable 
kind  of  curved  lines,  surfaces  and  solids,  is  generally  reducible  to 
the  calculations  of  b  and  c.  Add  t  and  you  have  the  means  of 
calculating  the  inconceivable  also. 

These  three  famous  ratios,  so  entirely  different  in  their  origin, 
and  so  utterly  incapable  of  exact  expression  in  number,  may  be 
connected  by  a  very  simple  bond.  If  we  multiply  1  by  2  ten 
times  we  obtain  1024 ;  and  if  we  multiply  1  by  1024  we  obtain 
the  same.  Since  ten  equal  multiplications  by  2  are  equal  to  one 
multiplication  by  1024 ;  we  may  say  that  multiplying  by  2  is  mul- 
tiplying by  1024  one-tenth  of  a  time ;  multiplying  by  8  is  multi- 
plying by  1024  three-tenths  of  a  time,  and  so  on.  Using  the  like  ex- 
pression we  may  say  that  b  multiplied  by  itself  half  c  times  gives 
4.8148.  Divide  1  by  this  number  and  we  get  .20788.  Next  we 
take  the  impossible,  inconceivable,  inexpressible  t,  and  multiply  it 
by  itself  »  times ;  and  can  readily  demonstrate  that  it  produces 
identically  that  same  .20788.  Here  then  are  the  three  famous  and 
inexpressible  ratios  of  numbers,  —  base,  circumference,  and  the 
imaginary,  —  bound  together  in  this  simple  law  ;  to  wit :  Multiply 
base  by  itself,  half  circumference  times ;  then  multiply  the  pro- 
duct by  imaginary,  imaginary  times;  and  the  final  product  is 
unity. 

The  object  of  this  long  discussion  of  such  abstruse  numerical 
relations  is  to  set  in  a  more  striking  light  the  marvellous  power  of 
the  human  body  as  an  unconscious  calculating  engine,  alluded  to 
in  more  than  one  of  the  preceding  chapters.  Number,  the  crea- 
tion of  conscious  spirit,  created  in  the  act  of  consciousness,  cannot, 
in  its  utmost  reaches  of  abstruseness,  go  beyond  the  power  of  the 
Infinite  Mind.  But  in  our  finite  thoughts  we  reach  conclusions, 
like  the  above,  binding  together  what  is  conceivable,  but  inex- 


98  GEOMETRY  AND  FAITH. 

plicable,  with  what  is  neither  conceivable  nor  expressible,  in  one 
inexplicable  bond. 

Yet  feeling,  or  emotion,  is  a  higher,  deeper  state  of  conscious- 
ness than  thought ;  and  often  carries  us  into  regions  of  the 
Divine  thonght  where  finite  thought  is  unable  to  follow.  In  some 
instances  the  reality  of  this  flight  into  higher  regions  has  afterward 
been  verified,  by  the  slower  laborious  ascent  of  the  finite  thought, 
in  paths  over  which  feeling  had  flown.  Thus  early  thinkers  de- 
clar.d  that  in  the  perception  of  musical  harmony,  the  soul  un- 
known to  herself  calculated  secretly  the  numerical  ratios  of  the 
undulations  of  the  air.  Modern  musical  statics  has  proved  that 
these  secret,  unconscious  numerical  calculations  by  the  musical 
ear,  by  the  mere  aesthetic  feelings,  have  been  far  more  numerous, 
complicated  and  precise  than  the  older  thinkers  had  supposed  ; 
and  that  elaborate  comparison,  calculation  and  experiment  demon- 
strate the  course  and  progress  of  music,  from  barbaric  times  to 
the  present,  to  have  been  in  perfect  agreement  in  all  respects 
with  laws  of  number,  not  revealed  until  very  recently.  It  is 
certainly  difficult  to  imagine  modes  by  which  experience  and 
habit  could  have  led  to  this  progress,  had  not  the  human  body 
been  in  its  very  creation  formed  in  exquisite  adaptation  to  those 
laws. 

I  have,  in  another  chapter,  alluded  to  Hay's  law  that  beauty  of 
geometric  form  depends  upon  the  division  of  the  right  angle  in 
harmonic  ratio.  Burke  argues  with  a  great  deal  of  misplaced  in- 
genuity to  prove  that  proportion  has  nothing  to  do  with  the  cause 
of  beauty ;  but  Hay  simply  takes  the  Parthenon,  the  Temple  of 
Theseus,  the  Lincoln  Cathedral,  and  other  universally  recognized 
types  of  beauty,  and  shows  that  their  actual  and  potential  angles  do 
always  stand  in  simple  harmonic  ratio  to  the  right  angle.  Here, 
therefore,  is  a  second  instance  in  which  the  secret  unconscious 
calculations  of  the  beholders  have  always  recognized  by  feeling 


NUMBER  AXD  PROPORTION.  99 

what  the  labors  of  Hay  have  shown  to  be  numerical  proportions 
in  architecture. 

But  I  think  I  have  proved  that  Hay's  law  is  incomplete ;  and 
that  the  eye  for  beauty  recognizes  sometimes  a  far  more  delicate 
and  involved  proportion  than  that  which  he  assumes.  He  illus- 
trates proportion  of  geometric  stationary  angle  by  the  proportion 
of  vibrations  in  time,  producing  a  musical  concord ;  and  draws  his 
law  from  an  examination  of  architectural  and  plastic  art.  But  the 
beauty  of  the  vegetable  world  led  me  to  consider  that  the  phyllo- 
tactic  ratio  fully  developed  in  the  preceding  chapter  might  also 
give  beauty  in  artificial  forms.  The  approximations  £,  ^,  f ,  and  f , 
would  be  also  included  in  Hay's  series.  To  test  this  question  I 
have  tried  various  experiments  with  satisfactory  results.  A  single 
one  will  show  their  nature.  I  drew  two  semi-ellipses ;  in  one,  the 
angle  (from  the  end  to  the  middle,  and  to  the  mid  side)  was  f ;  so 
that  either  by  phyllotaxy  or  by  harmony  it  should  be  beautiful. 
The  other  was  drawn  exactly  on  the  perfect  phyllotactic  angle ; 
which,  as  harmony,  would  be  rudely  discordant.  Showing  these 
two  ellipses,  without  explanation,  to  many  persons,  in  private  and 
in  schools,  and  taking  a  vote  on  the  merits  of  their  shape,  the  vote 
was  unanimous  on  their  both  being  good,  very  many  preferred 
the  phyllotactic  angle.  They  were  so  nearly  alike  in  their  propor- 
tions that  some  observers  mistook,  in  attempted  conscious  calcula- 
tion, and  thought  the  higher  one  flatter,  even  although  joining  in 
the  judgment  that  it  was  more  beautiful. 

How  much  more  wonderful,  if  possible,  is  that  unconscious  cal- 
culation of  numerical  relations  not  yet  confirmed  by  figures,  but 
proved  by  experiment  to  be  correctly  performed  in  higher  depart- 
ments of  art.  The  harmony  of  colors  unquestionably  depends  on 
numerical  relations  of  wave  lengths.  The  workman  at  a  Bessemer 
steel  factory  knows  when  to  draw  the  charge  by  a  secret  uncon- 
scious calculation  of  the  rapidity  of  vibrations,  millions  in  a 


100  GEOMETRY  AND  FAITH. 

second,  taking  place  in  the  flame.  All  the  world  have  by  similar, 
but  more  delicate  computations,  confirmed  and  endorsed  Titian's 
judgment  in  the  harmony  of  colors.  What  is  a  great  painting 
but  the  conveyance  of  great  thoughts  and  feelings  from  the  painter 
to  the  beholder  through  medium  of  numerical  ratios  of  angles  in 
the  drawing  and  of  vibrations  in  the  coloring.  It  is  ratio ;  it  is 
the  logos,  the  incarnate  word  of  the  painter,  imitating  the  Al- 
mighty Logos  by  which  the  heavens  and  the  earth  were  made, 
and  by  which  the  painter's  hand  had  received  the  skill  to  make, 
the  beholder's  eye  the  skill  to  interpret,  the  imitation. 

The  same  questions  may  be  asked,  the  same  answer  must  be 
given,  concerning  the  higher  ends  of  music.  I  have  demonstrated 
by  hundreds  of  carefully  conducted  experiments  upon  hundreds 
of  persons,  sometimes  upon  large  classes  in  schools,  that  fully 
three-fourths  of  an  audience  receive  from  a  musical  composition 
the  very  same  moral  mood  or  tone  of  feeling  which  filled  the  com- 
poser at  the  time  of  composing  it.  The  closing  chorus  in  Beetho- 
ven's "  Mount  of  Olives,"  played  upon  an  organ,  without  words, 
gives  to  a  hearer  who  has  not  heard  it  before,  and  who  is  not  in- 
formed of  what  it  is,  immediately  and  irresistibly  an  impression 
of  solemn  awe,  of  inexpressible  majesty,  of  penitence,  yet  of  the 
peace  of  forgiveness ;  of  gratitude  for  forgiveness,  and  a  sense  of 
reconciliation  through  mediation.  All  this  is  accomplished  sim- 
ply through  rhythmic  -modulations  ;  but  not,  as  in  spoken  words, 
through  any  conventional  or  artificial  associations  with  them.  It 
is  the  direct  natural  language  in  which  Beethoven  utters  his  pro- 
foundest,  deepest  faith.  If  the  hearer  have  known  anything  con- 
cerning the  tenets  of  the  Christian  faith,  he  will  be  certain  that 
the  composer  was  a  Christian  believer,  uttering  through  those 
chords  a  thanksgiving,  in  the  name  of  all  worlds,  for  the  recon- 
ciliation of  the  world  through  the  cross.  What  explanation  can 
be  given  of  this  high  power  of  the  artist,  in  whatever  department, 


NUMBER  AND  PROPORTION.  101 

to  pierce  direct  to  the  heart  without  the  aid  of  conscious  intellec- 
tion in  the  head  ?  I  see  none,  except  to  admit  that  given  in 
Emerson's  "  Problem,"  namely,  to  admit  that  the  forces  of  nature, 
partially  under  our  conscious  guidance,  are  wholly  under  the  con- 
trol of  that  same  infinite  wisdom  and  love  which  inspires  the 
artist,  and  interprets  his  work  to  his  audience. 


XVI. 

THE  DEVELOPMENT  OF  FORMS. 

IT  is  sometimes  said  that  classification  always  proceeds  by  a 
process  of  dismissing  from  attention  that  which  is  peculiar  to  the 
individuals  ;  and  fastening  the  mind  upon  some  arbitrarily  chosen 
points  of  resemblance.  Examples  of  this  method  are  found  in 
Linne's  artificial  system  of  botany,  and  in  Agassiz's  proposed 
classification  of  fish  by  their  scales.  All  classification  does  not, 
however,  proceed  upon  this  method.  Again,  it  is  said  that  the 
classification  of  the  organic  kingdoms  is  an  arrangement  accord- 
ing to  the  degree  of  development;  that  development  has  been  a 
continuous  process ;  and  the  degrees  have  been  accidentally,  or 
else  arbitrarily  marked  out.  A  little  examination  will  show  that 
this  view,  also,  is  erroneous. 

The  classification  of  the  organic  kingdoms  is  unquestionably 
based  upon  form,  in  the  geometric  sense  of  the  word.  An  accurate 
drawing,  an  accurate  plaster-cast,  the  impression  left  in  a  rock,  — 
these  are  sufficient  data  for  a  naturalist  to  decide  the  identity  of  a 
species.  If,  therefore,  classification  is  built  upon  development,  it 
is  upon  the  development  of  forms  in  space.  All  these  forms  have 
a  certain  amount  of  symmetry ;  their  outlines  and  surfaces  corre- 
spond, more  or  less  perfectly,  to  the  geometric  law.  We  have 
already  shown  that  this  conformity  to  law  is  obedience  to  thought. 
But  not  only  does  the  form  of  each  individual  conform  to  law  ; 
the  series  of  forms,  also,  as  Agassiz  has  shown,  indicates  a  law 
pervading  the  series ;  and  this  fact  is  inconsistent  with  the  opin- 
ion that  evolution,  if  there  has  been  evolution,  has  proceeded  con- 
tinuously, by  insensible  gradations. 
102 


THE  DEVELOPMENT  OF  FORMS.  103 

All  scientific  men,  at  the  present  day,  admit  the  reign  of  law  in 
organic  nature ;  even  those  who  believe  in  continuous  develop- 
ment. The  origin  of  the  universal,  invariable  law  of  inorganic 
nature  cannot,  however,  be  discovered  by  an  investigation  con- 
fined to  nature  herself.  That  question  lies  outside  the  realm  of 
science.  The  scientific  man  may  consider  it,  and  may  answer  it, 
wisely  or  unwisely ;  but  in  thus  doing  he  has  left  the  domain  of 
science,  and  entered  that  of  philosophy.  Science  infers  the  uni- 
versality and  invariability  of  natural  law,  by  a  "  simple  enumer- 
ation "  of  the  increasing  number  of  observed  phenomena  con- 
forming to  law;  philosophy  takes  a  firmer  ground.  She  con- 
ceives the  universality  of  law  to  flow  as  a  necessary  consequence 
from  the  grandest  and  most  certain  conclusion  of  human  thought ; 
the  being  of  an  infinitely  wise  Creator.  The  infinite  Wisdom 
foresaw  from  eternity  the  best  possible  modes  of  action,  and 
adopted  them.  No  occasion  can  arise  to  make  Infinite  Wisdom 
change  its  plan.  "  By  the  determinate  counsel  of  the  Lord,"  says 
the  wise  Hebrew,  "  were  his  works  from  the  beginning ; "  "  He 
gave  eternal  order  to  his  works,  and  to  the  atoms  thereof,  for  all 
generations ;  "  "  nor  to  eternity  shall  they  disobey  His  word."  No 
scientific  writer  of  the  nineteenth  century  can  more  distinctly 
affirm  the  universality  and  invariability  of  law  than  this  religious 
teacher  more  than  twenty  centuries  ago.  Historically,  as  well  as 
philosophically,  the  doctrine  is  the  direct  outcome  of  theistic  faith. 
To  one  whose  mind  has  grasped  the  conception  of  the  existence 
of  an  infinite  Deity,  of  infinite  wisdom  and  power,  the  whole 
universe  becomes  the  expression  of  a  single  Divine  Thought, 
almost  infinitely  full  of  detail,  and  thus  giving  endless  occupation 
to  our  finite  intellect ;  but  also  possessing  perfect  unity,  and  thus 
flooding  us  with  the  fulness  of  its  beauty.  In  this  theological 
form  the  doctrine  of  the  correlation  of  forces,  and  interdependence 
of  all  sciences,  was  familiar  to  metaphysical  and  theological  writ- 


104  GEOMETRY  AND  FAITH. 

ers,  long  before  its  tardy  confirmation  by  the  induction  of 
physicists.  The  natural  sciences,  like  the  mathematics,  have  their 
postulates ;  among  them  is  that  of  the  invariability  of  law ;  they 
need  also  the  postulate  of  the  universality  of  law.  The  doctrine 
of  a  continuous  development  will  be  found  difficult  to  reconcile 
with  either. 

This  doctrine  is  a  virtual  denial  of  the  existence  of  law  in  a 
department  in  which  the  whole  history  of  science  would  lead  us 
most  surely  to  expect  the  discovery  of  law.  Evolution  itself  has 
strong  antecedent  probabilities  in  its  favor  ;  it  is  only  against  the 
mode  of  evolution  by  continuous  change  and  accidental  arrest,  at 
certain  accidentally  determined  stages,  that  the  present  objections 
lie.  The  whole  course  of  scientific  study  up  to  the  middle  of  this 
century  had  revealed  more  and  more  clearly  the  presence  of  law, 
governing  the  atoms,  and  molecules,  and  masses  of  the  inorganic 
world.  In  order  to  do  this,  it  had  called  in  the  aid  of  the  mathe- 
matician. By  his  technical  language,  alone,  can  any  problem  of 
time  and  space,  matter  and  motion  be  exactly  and  definitely 
solved.  Even  in  the  organic  world,  he  had  begun  to  render 
the  botanist  and  zoologist  important  services.  He  had  shown 
that,  in  the  plant,  there  is  a  law  of  extreme  and  mean  ratio ;  that 
this  is  the  nearest  approach  to  a  uniform  distribution  of  the  leaves; 
he  had  shown  that,  in  the  zoologist's  classification  of  animals,  the 
first  great  division  rests  upon  laws  of  mechanical  equilibrium  in 
the  embryo,  as  imperious  as  the  law  of  an  arch.  The  natural  ex- 
pectation was  that  mathematical  bases  would  in  like  manner  be 
found  for  the  division  of  classes,  orders,  families  and  genera,  if  not 
the  species. 

But  under  the  theory  of  continuous  gradual  development  this 
course  of  scientific  progress  seems  arrested.  From  the  fact  that 
between  any  two  forms  of  nature  we  can  always,  if  we  have 
numerous  specimens,  find  intervening  links,  it  is  assumed  that 


THE  DEVELOPMENT  OF  FORMS.  105 

there  is  no  real  line  of  demarcation  between  the  forms.  There  is 
a  double  fallacy  in  this  assumption.  Between  any  chestnut  and 
any  oak,  for  example,  it  may  be  possible  to  find  an  intermediate 
form.  But  it  does  not  follow  that  it  would  be  possible  to  find  a 
form  of  which  a  thoroughly  trained  botanist  would  say,  '*  This  is 
either  an  oak  or  a  chestnut,  but  I  cannot  tell  which."  It  may 
easily  be  true  that  no  definition,  in  words,  of  an  oak  would  ex- 
clude every  chestnut ;  nor  any  definition  of  a  chestnut  exclude 
every  oak.  It  does  not  follow  that  the  senses  would  not  distin- 
guish ;  a  sharp  observer  sees  many  differences  not  easily  to  be 
described  in  words ;  as  those  between  an  oak  and  a  chestnut,  an 
apple  and  a  pear,  a  plum  and  a  cherry.  The  history  of  science  is 
full  of  instances  in  which  a  more  acute  observer  has  learned  to  see 
and  point  out  distinctions  between  things  which  had  been  con- 
founded. And,  secondly,  if  it  were  possible  to  show  that  between 
two  groups  the  transition  is  so  gradual  that  no  eye  can  detect  the 
dividing  line,  it  would  not  follow  that  no  dividing  line  existed, 
nor  that  the  groups  had  a  common  origin.  The  intellect  can  some- 
times clearly  distinguish  things,  indistinguishable  by  sense.  The 
elastic  curve  has,  at  the  extremities  of  its  series  of  variations,  a 
straight  line  and  a  circle  ;  indistinguishable  by  the  eye,  or  by  the 
imagination,  from  those  produced  by  varying  the  eccentricity  of 
the  ellipse.  Yet  reason,  transcending  imagination,  shows  them  to 
be  entirely  different ;  they  cannot  be  made  alike,  except  by  a 
process  which  would  confound  all  intellectual  distinctions,  and  de- 
stroy the  possibility  of  science. 

The  ellipse  and  the  elastic  curve  belong  to  genera  further  re- 
moved from  each  other  than  the  oak  and  the  chestnut ;  yet  each 
may  pass  by  variation  into  the  form  of  a  circle.  Similar  instances 
abound  in  the  curves  investigated  by  geometry.  An  observer 
unacquainted  with  mathematics  might  think  it  easy  to  pass  the 
elastic  curve  into  the  form  of  a  circle,  and  then  elongate  it  to  an 


106  GEOMETRY  AND  FAITH. 

ellipse ;  or  easy  to  frame  a  definition  of  the  one  curve,  which  should 
also  include  the  other ;  the  geometer  knows  that  neither  is  possi- 
ble. When  the  forms  of  the  chestnut  and  the  oak  are  as  thor- 
oughly understood  by  the  botanist,  as  those  of  the  ellipse  and  the 
elastic  curve  are  by  the  geometer,  he  will  probably  wonder  that 
the  two  genera  were  ever  considered  difficult  to  define  and  separate. 
It  may  even  be  that  the  mathematician  will  demonstrate  the  dif- 
ference of  the  forms.  It  is  the  mathematician  alone  who  has  in- 
troduced precision  and  certainty  into  the  other  physical  sciences ; 
and  he  will  probably,  at  some  day,  introduce  them  into  biology. 
The  botanist  and  the  zoologist  may  rebel,  but  they  will  rebel  in 
vain.  The  numbers  of  Pythagoras  and  the  axioms  of  Euclid  are 
inexorable.  The  fates  themselves  cannot  violate  the  laws  of  geom- 
etry, arithmetic,  and  algebra ;  much  less  can  fluttering  theorists 
break  through  those  adamantine  bars. 

The  erratic  genius  of  De  Maillet,  and  of  Erasmus  Darwin,  has 
built  an  ingenious  theory,  perfected  under  the  magic  influence  of 
Charles  Darwin,  by  the  aid  of  a  host  of  writers,  which  flatters 
itself  that  it  explains  all  organic  forms  in  complete  independence 
of  geometrical  and  arithmetical  law.  It  makes  the  labors  of 
classification  as  empty  of  real  meaning  as  though  they  had  been 
expended  upon  the  forms  of  clouds  or  upon  the  disposition  of  the 
settlings  of  cups  of  tea.  This  theory  of  insensible,  accidental 
variation,  modified  and  arrested  by  the  surroundings,  is  a  virtual 
assertion  that  the  whole  problem  ©f  classification  is  a  delusion. 
Expand  the  theory  to  its  most  complete  form,  and  it  becomes  self- 
contradictory  and  self-destructive.  It  would  declare  all  personifi- 
cation of  objects,  and  all  entification  of  attributes,  mythical  and 
illusory,  and  make  monotheism  the  last  step  of  the  illusion  pre- 
vious to  awaking  and  discovering  that  it  is  all  a  dream.  Thus  its 
evolution  contains  in  itself  an  illogical  breach  of  continuity. 
When  a  mathematical  series  tends  through  an  indefinite  series  of 


THE  DEVELOPMENT  OF  FORMS.  107 

terms  toward  unity,  it  is  illogical  to  assume  either  that  it  ends  in 
zero  or  that  it  becomes  indefinite.  As  well  might  the  modern 
correlation  of  forces,  tending  constantly  to  show  that  all  phenom- 
ena are  modes  of  motion,  be  held  to  show  that  there  is  no  motion, 
or  at  least  that  we  can  not  know  that  there  is  motion,  as  the 
universal  tendency  of  cultivated  thought  to  reduce  all  super- 
natural forces  to  the  will  of  one  God  be  held  to  lead  properly  to 
atheism  or  to  agnosticism.  If  the  reduction  of  personification  in 
things  and  ideas  to  smaller  "and  smaller  numbers  logically  leads 
to  the  denial  of  real  being  in  any  entified  idea,  it  should  lead 
also  to  the  denial  of  personality  in  any  thing;  that  is,  to  the 
denial  of  personality  in  our  fellow-animals,  and  even  in  our  fel- 
low-men. In  fact,  this  theory  goes  further;  it  ignores  the  ex- 
stence  of  personality  in  one's  self ;  it  makes  the  theorist  himself 
a  non-existent  being,  knowing  only  that  one  thing,  that  he  does 
not  exist. 

The  students  of  botany  and  zoology  have  been  laboring  for 
nearly  twenty-five  centuries  in  one  direction,  with  steady  pro- 
gress ;  there  has  been  a  substantial  agreement  among  them  as  to 
the  proper  divisions  of  the  animal  and  vegetable  kingdoms ;  their 
divisions  have  been  concerning  the  grounds  of  the  division.  For 
the  final  settlement  of  the  questions  of  classification  the  aid  of 
the  mathematics  is  necessary.  This  has  been  the  destiny  of 
the  other  physical  sciences ;  it  is  that  of  biology  also.  There  is  no 
breach  of  continuity  in  nature ;  geometric  and  algebraic  law  rules 
absolute  over  all  that  can  be  bounded  in  space  and  time.  The 
vagueness  of  arbitrary  variation  and  survival  of  the  fittest  is  a 
poetical  dream ;  it  must  give  way  to  the  intellectual,  scientific 
sternness  of  invariable  law,  bounded  by  invariable  limits.  As  the 
four  primitive  forms  of  the  embryo  flow  from  necessary  mechani- 
cal conditions,  inflexible  as  the  law  of  equilibrium  in  arches ;  so 
the  classes,  orders,  families,  genera,  will  be  found  to  be  formed 


108  GEOMETRY  AND  FAITH. 

by  conditions  sharply  defined  in  nature  ;  and  hereafter  to  be, 
through  the  aid  of  mathematics,  sharply  defined  in  human 
thought. 

No  geometer  will  willingly  relinquish  his  hope  of  great  triumphs 
in  the  future ;  when  the  new  mathematical  methods  of  the  nine- 
teenth century  shall  have  been  as  faithfully  applied  to  the  prob- 
lems of  organic  form,  as  the  methods  of  the  seventeenth  century 
have  been  to  those  of  inorganic  matter.  If  the  melodious  wail  of 
Darwin's  bugle  leads  the  naturalists  to  retreat  from  the  grand 
problem  of  classification  ;  and  if  the  trumpet  of  the  elder  Agassiz 
fails  to  rally  them ;  the  mathematicians  will  press  forward  and 
gain  the  high  honor  of  victory,  on  the  noblest  field  of  natural  sci- 
ence. Mathematical  science  cannot  admit  the  possibility  that  the 
rhythm  and  symmetry  of  the  organic  kingdoms  is  an  accidental 
result  of  accidental  variations ;  there  must  be  algebraic  and  geo- 
metric law  at  the  basis,  not  only  of  each  organic  form,  but  of  the 
series  of  forms.  The  series  has  a  unity ;  capable,  when  men  have 
attained  a  fuller  comprehension  of  it,  of  expression  in  terms  of 
thought. 

The  rhythm  and  harmony  of  a  symphony  reveal  not  only  the 
skill  of  the  orchestra  and  its  conductor ;  but  the  great  mind  and 
noble  heart  of  the  composer.  The  rhythm  and  harmony  of  the 
organic  world  reveal  the  power,  the  wisdom,  and  the  love  of  God. 
So  long  as  man  is  less  than  the  universe,  his  wisest  and  best  course 
is  to  seek  everywhere,  not  for  discords  and  maladaptations,  but  for 
harmonies,  correlations,  adaptations.  The  universe  is  the  sum  of 
all  symmetries  ;  and  contains  all  geometries,  architectures,  sculpt- 
ures, and  pictorial  arts.  It  is  the  sum  of  all  rhythms,  melodic  or 
harmonic ;  and  contains  all  algebra,  poetry,  music  and  dance. 
The  Divine  Word,  which  created  it,  is  wisdom  and  love ;  and 
manifests  wisdom  and  love  in  every  syllable  and  tone  in  which 
it  utters  itself;  not  least  in  the  wondrous  series  of  the  forms 


THE  DEVELOPMENT  OF  FORMS.  109 

of  plants  and  animals;  swaying,  in  the  responsive  rhythm  of 
growth  and  decay,  sleep  and  activity,  generation  and  succession, 
to  the  periodic  march  of  the  earth,  the  moon,  the  planets  and 
the  sun. 


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